K11a32

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K11a31.gif

K11a31

K11a33.gif

K11a33

Contents

K11a32.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a32 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,17,21,18 X12,19,13,20 X10,22,11,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 14 2 16 18 6 22 20 12 10
A Braid Representative
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A Morse Link Presentation K11a32 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a32/ThurstonBennequinNumber
Hyperbolic Volume 16.0547
A-Polynomial See Data:K11a32/A-polynomial

[edit Notes for K11a32's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a32's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-14 t^2+32 t-41+32 t^{-1} -14 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 139, 2 }
Jones polynomial q^9-4 q^8+8 q^7-14 q^6+19 q^5-22 q^4+23 q^3-19 q^2+15 q-9+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +z^4 a^{-2} +7 z^4 a^{-4} -3 z^4 a^{-6} -z^4-z^2 a^{-2} +11 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -z^2- a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +5 z^9 a^{-3} +9 z^9 a^{-5} +4 z^9 a^{-7} +9 z^8 a^{-2} +20 z^8 a^{-4} +17 z^8 a^{-6} +6 z^8 a^{-8} +8 z^7 a^{-1} +9 z^7 a^{-3} +2 z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} -11 z^6 a^{-2} -45 z^6 a^{-4} -43 z^6 a^{-6} -12 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-12 z^5 a^{-1} -32 z^5 a^{-3} -40 z^5 a^{-5} -31 z^5 a^{-7} -10 z^5 a^{-9} +5 z^4 a^{-2} +37 z^4 a^{-4} +35 z^4 a^{-6} +6 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-a z^3+7 z^3 a^{-1} +26 z^3 a^{-3} +41 z^3 a^{-5} +31 z^3 a^{-7} +8 z^3 a^{-9} -3 z^2 a^{-2} -19 z^2 a^{-4} -16 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +2 z^2-2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -10 z a^{-7} -2 z a^{-9} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8}
The A2 invariant Data:K11a32/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a32/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (3, 4)
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
12 32 72 110 2 384 \frac{1472}{3} \frac{224}{3} 32 288 512 1320 24 \frac{21471}{10} \frac{3494}{15} \frac{2234}{5} \frac{43}{2} \frac{191}{10}

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 t^rq^j are shown, along with their alternating sums \chi (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 K11a32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          3 -3
15         51 4
13        93  -6
11       105   5
9      129    -3
7     1110     1
5    812      4
3   711       -4
1  39        6
-1 16         -5
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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.

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a31

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K11a33