# 7 1

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 7 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,6,-3,7,-4,1,-5,2,-6,3,-7,4/goTop.html 7_1's page] at Knotilus! Visit 7 1's page at the original Knot Atlas! 7_1 should perhaps be called "The Septafoil Knot", following the trefoil knot and the cinquefoil knot. See also T(7,2).

 Interlaced form of 7/2 star polygon or "septagram" Decorative interlaced form of 7/2 star polygon or "septagram" 3D depiction Heptagram of intersecting circles.

### Knot presentations

 Planar diagram presentation X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7 Gauss code -1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4 Dowker-Thistlethwaite code 8 10 12 14 2 4 6 Conway Notation [7]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 7, width is 2,

Braid index is 2

[{9, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]
 Knot 7_1. A graph, knot 7_1.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (unknown error\text): \text{$\$\$Failed} Hyperbolic Volume Not hyperbolic A-Polynomial See Data:7 1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 3 Topological 4 genus 3 Concordance genus ConcordanceGenus(Knot(7,1)) Rasmussen s-Invariant -6

### Polynomial invariants

 Alexander polynomial t3 + t−3−t2−t−2 + t + t−1−1 Conway polynomial z6 + 5z4 + 6z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 7, -6 } Jones polynomial −q−10 + q−9−q−8 + q−7−q−6 + q−5 + q−3 HOMFLY-PT polynomial (db, data sources) $a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6$ Kauffman polynomial (db, data sources) a13z + a12z2 + a11z3−a11z + a10z4−2a10z2 + a9z5−3a9z3 + a9z + a8z6−5a8z4 + 7a8z2−3a8 + a7z5−4a7z3 + 3a7z + a6z6−6a6z4 + 10a6z2−4a6 The A2 invariant −q30−q28−q26 + q18 + q16 + 2q14 + q12 + q10 The G2 invariant q168−q136−q134−q128−q126−q124−q118−q116−q102−q96−q94−q92 + q88−2q84 + 2q80 + q78 + q72 + 3q70 + 2q68 + q64 + 2q62 + 2q60 + q58 + q54 + q52 + q50

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (6, -14)
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 24 −112 288 684 100 −2688 $-\frac{13888}{3}$ $-\frac{2464}{3}$ −560 2304 6272 16416 2400 $\frac{160231}{5}$ $\frac{21548}{15}$ $\frac{58148}{5}$ 163 $\frac{7351}{5}$

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 = -6 is the signature of 7 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-7-6-5-4-3-2-10χ
-5       11
-7       11
-9     1  1
-11        0
-13   11   0
-15        0
-17 11     0
-19        0
-211       -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −7 i = −5 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}$ r = −2 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −1 r = 0 ${\mathbb Z}$ ${\mathbb Z}$

### 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).

Back to the top.