K11a161

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K11a160.gif

K11a160

K11a162.gif

K11a162

Contents

K11a161.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a161 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X14,8,15,7 X12,10,13,9 X2,11,3,12 X8,14,9,13 X20,15,21,16 X22,17,1,18 X6,19,7,20 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -4, 8, -11, 9, -3, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 18 14 12 2 8 20 22 6 16
A Braid Representative
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A Morse Link Presentation K11a161 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a161/ThurstonBennequinNumber
Hyperbolic Volume 11.0126
A-Polynomial See Data:K11a161/A-polynomial

[edit Notes for K11a161's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a161's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+8 t^2-12 t+13-12 t^{-1} +8 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, -4 }
Jones polynomial q^2-2 q+4-6 q^{-1} +7 q^{-2} -8 q^{-3} +9 q^{-4} -7 q^{-5} +6 q^{-6} -4 q^{-7} +2 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) -z^2 a^8-2 a^8+2 z^4 a^6+6 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4+a^4-z^6 a^2-4 z^4 a^2-5 z^2 a^2-3 a^2+z^4+3 z^2+2
Kauffman polynomial (db, data sources) z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+3 z^5 a^9-2 z^3 a^9+z a^9+4 z^6 a^8-6 z^4 a^8+5 z^2 a^8-2 a^8+4 z^7 a^7-6 z^5 a^7-z^3 a^7+2 z a^7+4 z^8 a^6-9 z^6 a^6+z^4 a^6+4 z^2 a^6-3 a^6+3 z^9 a^5-8 z^7 a^5+3 z^5 a^5-2 z^3 a^5+2 z a^5+z^{10} a^4+2 z^8 a^4-21 z^6 a^4+29 z^4 a^4-13 z^2 a^4+a^4+5 z^9 a^3-23 z^7 a^3+31 z^5 a^3-16 z^3 a^3+5 z a^3+z^{10} a^2-z^8 a^2-14 z^6 a^2+32 z^4 a^2-20 z^2 a^2+3 a^2+2 z^9 a-11 z^7 a+19 z^5 a-12 z^3 a+3 z a+z^8-6 z^6+12 z^4-9 z^2+2
The A2 invariant Data:K11a161/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a161/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_14, K11n2,}

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

Vassiliev invariants

V2 and V3: (2, -7)
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 -56 32 \frac{748}{3} \frac{140}{3} -448 -\frac{3536}{3} -\frac{512}{3} -216 \frac{256}{3} 1568 \frac{5984}{3} \frac{1120}{3} \frac{97111}{15} -\frac{668}{5} \frac{128644}{45} \frac{1049}{9} \frac{5431}{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 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=-4 is the signature of K11a161. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
5           11
3          1 -1
1         31 2
-1        31  -2
-3       43   1
-5      54    -1
-7     43     1
-9    35      2
-11   34       -1
-13  13        2
-15 13         -2
-17 1          1
-191           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\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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K11a160.gif

K11a160

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K11a162