# 7 4

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-5,6,-7,2,-1,3,-4,7,-6,5,-2,4,-3/goTop.html 7_4's page] at Knotilus! Visit 7 4's page at the original Knot Atlas! Simplest version of Endless knot symbol.
 Celtic or pseudo-Celtic knot a knot seen at the Castle of Kornik [1] Susan Williams' medallion [2], the "Endless knot" of Buddhism [3] Ornamental "Endless knot" A 7-4 knot reduced from TakaraMusubi with 9 crossings [4] TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5] Buddhist Endless Knot Ornamental Endless Knot Albrecht Dürer knot, 16th-century A laser cut by Tom Longtin [6] Unicursal hexagram of occultism

### Knot presentations

 Planar diagram presentation X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9 Gauss code 1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3 Dowker-Thistlethwaite code 6 10 12 14 4 2 8 Conway Notation [313]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

[{3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {1, 8}, {9, 2}, {4, 1}]
 Knot 7_4. A graph, knot 7_4.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 1 Bridge index 2 Super bridge index {3,4} Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (unknown error\text): \text{$\$\$Failed} Hyperbolic Volume 5.13794 A-Polynomial See Data:7 4/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 1 Topological 4 genus 1 Concordance genus ConcordanceGenus(Knot(7,4)) Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial 4t + 4t−1−7 Conway polynomial 4z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 15, 2 } Jones polynomial −q8 + q7−2q6 + 3q5−2q4 + 3q3−2q2 + q HOMFLY-PT polynomial (db, data sources) −a−8 + z2a−6 + 2z2a−4 + 2a−4 + z2a−2 Kauffman polynomial (db, data sources) z6a−6 + z6a−8 + 2z5a−5 + 3z5a−7 + z5a−9 + 3z4a−4−3z4a−8 + 2z3a−3−2z3a−5−8z3a−7−4z3a−9 + z2a−2−4z2a−4−3z2a−6 + 2z2a−8 + 4za−7 + 4za−9 + 2a−4−a−8 The A2 invariant q−2−q−4 + q−8 + q−10 + 2q−12 + q−14 + q−16−q−20−q−24−q−26 The G2 invariant q−10−q−12 + q−14−q−16−q−22 + 4q−24−2q−26 + 2q−28−q−30 + q−34−2q−36 + 3q−38−2q−40 + q−44 + 2q−48 + q−50−q−52 + 2q−54−q−56 + 3q−58 + q−60−2q−62 + 6q−64−3q−66 + 4q−68 + 2q−70−3q−72 + 4q−74−3q−76 + 3q−78−q−80−q−82 + q−84−2q−86 + 2q−88−3q−92−q−94−2q−96−4q−102 + 2q−104−3q−106 + q−108 + q−110−4q−112 + 3q−114−2q−116 + q−118−2q−122 + 2q−124 + q−128

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_2,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

### Vassiliev invariants

 V2 and V3: (4, 8)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 16 64 128 $\frac{1016}{3}$ $\frac{184}{3}$ 1024 $\frac{5824}{3}$ $\frac{1024}{3}$ 320 $\frac{2048}{3}$ 2048 $\frac{16256}{3}$ $\frac{2944}{3}$ $\frac{168062}{15}$ $-\frac{1176}{5}$ $\frac{233288}{45}$ $\frac{898}{9}$ $\frac{11102}{15}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### Khovanov Homology

 The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 2 is the signature of 7 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
01234567χ
17       1-1
15        0
13     21 -1
11    1   1
9   12   1
7  21    1
5  1     1
312      -1
11       1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 r = 0 ${\mathbb Z}$ ${\mathbb Z}$ r = 1 ${\mathbb Z}^{2}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 5 ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 6 ${\mathbb Z}$ r = 7 ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session, or any of the Computer Talk sections above.

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

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