# 6 1

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 6 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6_1's page at Knotilus! Visit 6 1's page at the original Knot Atlas! 6_1 is also known as "Stevedore's Knot" (see e.g. [1]), and as the pretzel knot P(5,-1,-1).

 A Kolam of a 3x3 dot array 3D depiction

### Knot presentations

 Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X5,12,6,1 X11,6,12,7 Gauss code -1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5 Dowker-Thistlethwaite code 4 8 12 10 2 6 Conway Notation [42]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 7, width is 4,

Braid index is 4

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index {3,4} Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-3] Hyperbolic Volume 3.16396 A-Polynomial See Data:6 1/A-polynomial

[edit Notes for 6 1's three dimensional invariants]
6_1 is a ribbon knot (drawings by Yoko Mizuma):

 a ribbon diagram isotopy to a ribbon
6_1 is doubly slice, by Scott Carter
Scott Carter notes that 6_1 is doubly slice - it bounds two distinct slice disks. He says: "this was spoken of in Fox's Example 10, 11, and 12 in a Quick Trip through Knot Theory ... BTW, the cover of Carter and Saito's Knotted Surfaces and Their Diagrams contains an illustration of such a slice disk". A picture is on the right.

### Four dimensional invariants

 Smooth 4 genus 0 Topological 4 genus 0 Concordance genus 0 Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial −2t + 5−2t−1 Conway polynomial 1−2z2 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 9, 0 } Jones polynomial q2−q + 2−2q−1 + q−2−q−3 + q−4 HOMFLY-PT polynomial (db, data sources) a4−z2a2−a2−z2 + a−2 Kauffman polynomial (db, data sources) a3z5 + az5 + a4z4 + 2a2z4 + z4−3a3z3−2az3 + z3a−1−3a4z2−4a2z2 + z2a−2 + 2a3z + 2az + a4 + a2−a−2 The A2 invariant q14 + q12−q6−q4 + q−2 + q−6 + q−8 The G2 invariant q66 + q62−q60 + q56−q54 + 2q52 + q46 + q42−q38 + q32−2q28 + q26 + q24−2q20−2q18 + q16−q14 + q12−2q10−q8 + 2q6−q4−1 + q−4 + q−10 + 2q−14−q−18 + q−20 + q−24 + q−28 + q−34 + q−38

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

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

### Vassiliev invariants

 V2 and V3: (-2, 1)
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 −8 8 32 $\frac{116}{3}$ $\frac{52}{3}$ −64 $-\frac{304}{3}$ $-\frac{64}{3}$ −24 $-\frac{256}{3}$ 32 $-\frac{928}{3}$ $-\frac{416}{3}$ $-\frac{2791}{15}$ $\frac{884}{15}$ $-\frac{10084}{45}$ $\frac{343}{9}$ $-\frac{871}{15}$

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 = 0 is the signature of 6 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-1012χ
5      11
3       0
1    21 1
-1   11  0
-3   1   -1
-5 11    0
-7       0
-91      1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −4 ${\mathbb Z}$ r = −3 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}$ r = 2 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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