K11n7

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K11n6.gif

K11n6

K11n8.gif

K11n8

Contents

K11n7.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n7 at Knotilus!


Knot K11n7.
A graph, knot K11n7.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X18,12,19,11 X13,6,14,7 X20,16,21,15 X12,18,13,17 X22,20,1,19 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, 6, -9, -7, 4, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code 4 8 10 -14 2 18 -6 20 12 22 16
A Braid Representative
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A Morse Link Presentation K11n7 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n7/ThurstonBennequinNumber
Hyperbolic Volume 13.8902
A-Polynomial See Data:K11n7/A-polynomial

[edit Notes for K11n7'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 K11n7's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+16 t-21+16 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial -2 q^6+5 q^5-8 q^4+11 q^3-11 q^2+11 q-9+6 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+4 z^2 a^{-2} -4 z^2+a^2+2 a^{-2} + a^{-4} - a^{-6} -2
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +6 z^8 a^{-2} +3 z^8 a^{-4} +3 z^8+3 a z^7+7 z^7 a^{-1} +7 z^7 a^{-3} +3 z^7 a^{-5} +a^2 z^6-7 z^6 a^{-2} -z^6 a^{-4} +z^6 a^{-6} -4 z^6-9 a z^5-24 z^5 a^{-1} -17 z^5 a^{-3} -2 z^5 a^{-5} -3 a^2 z^4-9 z^4 a^{-2} -z^4 a^{-4} +4 z^4 a^{-6} -7 z^4+8 a z^3+17 z^3 a^{-1} +11 z^3 a^{-3} +5 z^3 a^{-5} +3 z^3 a^{-7} +3 a^2 z^2+9 z^2 a^{-2} -3 z^2 a^{-6} +9 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -3 z a^{-5} -2 z a^{-7} -a^2-2 a^{-2} + a^{-4} + a^{-6} -2
The A2 invariant Data:K11n7/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n7/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n131, K11n160,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n36, K11n44,}

Vassiliev invariants

V2 and V3: (1, 3)
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
4 24 8 \frac{158}{3} \frac{10}{3} 96 208 0 56 \frac{32}{3} 288 \frac{632}{3} \frac{40}{3} \frac{26671}{30} -\frac{1222}{15} \frac{18902}{45} \frac{305}{18} \frac{1711}{30}

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 K11n7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         2-2
11        3 3
9       52 -3
7      63  3
5     55   0
3    66    0
1   46     2
-1  25      -3
-3 14       3
-5 2        -2
-71         1
Integral Khovanov Homology

(db, data source)

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

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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K11n6.gif

K11n6

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K11n8