### Gauss Diagram Invariants for Knots and Links

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We begin this paper by reviewing the definition virtual knots and links as well as the definition of flat and free knots. We discuss the correspondence between virtual knots, Gauss codes and Gauss diagrams. Next, we construct a knot invariant that is based on a linking number. In section 3 , we define parity and discuss its relation to the linking number. This invariant does not appear to be related to either the Jones polynomial or the Alexander polynomial.

Based on their construction, these families of invariants are related to the Kauffman finite type invariants in [ 4 ]. A virtual link diagram is a decorated immersion of n copies of S 1 into the plane with two types of crossings: virtual and classical. Two virtual link diagrams are equivalent if they are related by a finite sequence of the extended Reidemeister moves which are shown in figure 1.

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Virtual links are equivalence classes of virtual link diagrams. We use the term knot to indicate a link with one component. Flat virtual link diagrams provide no information about which edge overpasses or underpasses at each crossing. Flat crossings are indicated by a solid X and the virtual crossings are indicated by a circled double points. Correspondingly, flat virtual links are equivalence classes of flat virtual diagrams determined by the flat Reidemeister moves.

### Identifying knots and links

Equivalence classes of flat virtual knots are in one to one correspondence with the homotopy classes of virtual knots Reidemeister moves and self-crossing change. This is not true for links, since homotopy classes of links do not permit crossing changes between different components. Free knots and links are equivalence classes of flat diagrams determined by the set of extended flat Reidemeister moves and the flat virtualization moves shown in figure 2. We study oriented versions of these types of links. Given an orientation of the link components, a classical crossing in a virtual link diagram has either a positive or negative sign.

## Smoothed Invariants

The convention determining the sign of the crossing is illustrated in figure 3. The writhe of a link K is denoted w and is the sum of the crossings signs:. A long virtual link is an immersion of n copies of the unit interval into the plane. Equivalence classes of long virtual links are determined by the extended Reidemeister moves, but these local moves cannot pass throught the endpoints of the unit interval.

For flat links and free links, we have analogous versions of long virtual links. In a long link, we order the components and orient each unit interval downwards. A decorated Gauss code and a corresponding Gauss diagram can be obtained from a virtual link. To construct the Gauss code, choose an ordering of the link components, an orientation and basepoint for each link component.

## Knot theory

Label each crossing in the diagram with a symbol. For each component, we construct a word as follows. Traverse the component in the direction of orientation and record the information about the crossing as shown in figure 4. Overcrossing strands are marked with an O. Similarly, a negative crosssing is recorded as a minus.

We show the Gauss code and diagram corresponding to the left handed trefoil in figure 4. Note that the Gauss code of a knot consists of a single word whereas Gauss code for an n component link consists of n words and is referred to as a Gauss phrase. We can express the Reidemeister moves as the following changes in the Gauss code. Information about the crossing sign and position of the arrow head was deleted for convenience. Reidemeister 2: Insertion Deletion of b c. Reidemeister 3: The subsections a b c b a c switched to b a b c c a.

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Gauss codes, modulo the changes in the code given by items and cyclic permutation of the symbols in a word, and the ordering of the words are in one to one correspondence to virtual link diagrams. From the Gauss code, we construct the corresponding Gauss diagram. For each component construct a circle. Label each circle following a clockwise orientation with the symbols in the word corresponding to the component. Connect each instance of the symbol with a chord. Place an arrowhead on the chord pointing to the symbol corresponding to the overpass.

Mark each chord with the sign of the crossing. Equivalence classes of Gauss diagrams are in one to one correspondence with virtual links. We can construct equivalence classes of Gauss codes that are in one to one correspondence with flat virtual links. Flat crossings do not have a positive or negative orientation. We eliminate the crossing sign information and use the arrow on the chord diagram to determine a local framing. The local framing convention is shown in figure 5.

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Reversing the direction of the arrow on chord corresponds to virtualizing the crossing as shown in figure 2. Gauss codes equivalent to free knots are constructed by removing both the sign information and the arrow from the chord. The Gauss diagram equivalents of the Reidemeister moves for free knots are shown in figure 6. Let L be an oriented virtual link diagram with components A and B. We define. We can also define a virtual linking number.

For each virtual crossing that involves both the A and B components, transform the virtual crossing into a classical crossing by designating A as the overpassing strand and B as the underpassing strand. Let V denote this set of crossings. Several contributions are of an Chemistry of the Textiles Industry. The manufacture and processing of textiles is a complex and essential industry requiring many diverse The manufacture and processing of textiles is a complex and essential industry requiring many diverse skills to ensure profitability.

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A study of cosmic ray spectra, time variations, abundances, gradients, and anisotropy provides a wealth of data Conservation Laws in Variational Thermo-Hydrodynamics. This study is one of the first attempts to bridge the theoretical models of variational This study is one of the first attempts to bridge the theoretical models of variational dynamics of perfect fluids and some practical approaches worked out in chemical and mechanical engineering in the field newly called thermo-hydrodynamics. In recent years, applied Using this definition, we say that a crossing is even respectively odd if the corresponding chord in the Gauss diagram is even odd.

In figure 9 , the red chord has odd parity. We define oriented parity. In a Gauss diagram, each chord is assigned an integer value. We consider the intersection of two chords, t and s. Chord t intersects chord s positively if we encounter the head of t when we traverse the circle in the clockwise direction from the tail of s to its head.

Otherwise, t intersects s negatively and we encounter the tail of t as the circle is traversed from the tail of s to its head. To compute the oriented parity of a chord, we sum the signs of the chords that intersect is positively p and sum the signs of the chords that intersect it negatively n. The oriented parity is:. In figure 9 , the the sum of the signs of the two chords that intersect negatively sum to zero and the sign of the three chords that intersect positively sum to three.

Hence, the oriented parity of the chord in our example is three. Otherwise, it is odd. The oriented parity is sometimes referred to as the index. By abuse of notation, the terms even and odd are used interchangably with zero and one. Manturov describes further generalizations of parity in [ 5 ]. Manturov has also investigated parity in [ 6 ]. The virtual figure eight knot is shown in figure We obtain three vertically smoothed states.

The virtual knot shown in figure 14 is knot 4. Hence, the smoothed invariants differentiate this knot from the unknot while the Jones polynomial does not. Modulo two, these matrices are degree one Kauffman finite type invariants [ 3 ]. Additionally, these invariants are always zero matrices for classical knots. We can construct an invariant of flat virtual knots that is a formal sum of flat knots. Each flat knot in the sum is obtained by vertically smoothing pairs of crossings that correspond to intersticed symbols with opposite parity and flattening the crossings.

Let P denote the set of pairs of crossings that correspond to pairs of chords that intersect and have opposite parity. Let K p denote the virtual knot diagram obtained by smoothing such a pair of crossings.

Proof: Chords corresponding to a Reidemeister I move do not intersect any other chord. There is no contribution to the formal sum. Chords corresponding to a Reidmeister II move have the same parity. Hence any contribution to the formal sum is determined by a crossing from the Reidmeister II move and an external chord. However, the other chord from the from the Reidemeister move also intersects the external chord.

Both smoothings result in the same flat diagram and the net contribution is zero. In the Reidemeister III move, we have two cases: two internal chords or an internal chord and an external chord. For the case of two internal chords, smoothing the three crossings in the Reidemeister III individually produces equivalent two sets of flat diagrams that are equivalent.

Then, smoothing an external crossing in addition to the internal crossing, produces equivalent sets of flat knots. Hence, both sides of the Reidemeister III move make the same contribution to the formal sum. The case of two internal crossings requires more analysis. The parity of a chord crossing does not change under the Reidemeister moves. However, the chords that intersect do change.

We consider the case shown in figure There, chord 3 has odd parity and chords 1 and 2 have even parity. On the right hand side, the chords do not intersect and no diagrams can be obtained from a smoothing pair involving two internal crossings. On the left hand side, we smooth the following pairs: 1,2 and 1,3.

Smoothing these pairs results in two equivalent flat diagrams. Hence, modulo two, the net contribution to the formal sum is zero. To finish the proof, we check all possible parity and crossing combinations from the Gauss diagrams corresponding to the Reidemeister III moves.

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## Polynomials and Homotopy of Virtual Knot Diagrams

This formal sum may also be zero. We realize that the chords involved make no contribution to the formal sum. We consider a diagram obtained by smoothing one from a Reidemeister three move and a crossing external to move. However, both sides of the Reidemeister III moves produce homotopic sets of diagrams. For a diagram obtained by smoothing two crossings internal to the Reidemeister III diagram, both sides of the move produce equivalent sets of flat virtual diagrams under this smoothing. Proof: Free knots are equivalence clases of diagrams determined by the set of flat extended Reidemeister moves and the virtualization move.

In the Gauss diagram, virtualization corresponds to reversing the direction of the on arrow on the chord. Virtualization does not change the parity of the chord. As a result, it does not change the elements of the set P. The summands are either equivalent flat diagrams or diagrams related by virtualizing the crossing c. It should be apparent that smoothing the pairs 1 , 2 and 1 , 3 produces equivalent diagrams, so we obtain the sum of the diagrams obtained by smoothing 4 , 6 and 5 , 6 respectively.