K11a117

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K11a116.gif

K11a116

K11a118.gif

K11a118

Contents

K11a117.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a117 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X18,8,19,7 X2,10,3,9 X22,11,1,12 X8,14,9,13 X20,16,21,15 X6,18,7,17 X16,20,17,19 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -3, 8, -10, 9, -4, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 10 14 18 2 22 8 20 6 16 12
A Braid Representative
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A Morse Link Presentation K11a117 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a117/ThurstonBennequinNumber
Hyperbolic Volume 14.597
A-Polynomial See Data:K11a117/A-polynomial

[edit Notes for K11a117'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 K11a117's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-27 t+35-27 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 117, 4 }
Jones polynomial -q^{11}+4 q^{10}-8 q^9+12 q^8-17 q^7+19 q^6-18 q^5+16 q^4-11 q^3+7 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} +2 z^4 a^{-8} +2 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-10} + a^{-2} + a^{-6} - a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +7 z^9 a^{-7} +4 z^9 a^{-9} +4 z^8 a^{-4} +9 z^8 a^{-6} +12 z^8 a^{-8} +7 z^8 a^{-10} +3 z^7 a^{-3} +z^7 a^{-5} -5 z^7 a^{-7} +4 z^7 a^{-9} +7 z^7 a^{-11} +z^6 a^{-2} -7 z^6 a^{-4} -22 z^6 a^{-6} -27 z^6 a^{-8} -9 z^6 a^{-10} +4 z^6 a^{-12} -8 z^5 a^{-3} -14 z^5 a^{-5} -11 z^5 a^{-7} -18 z^5 a^{-9} -12 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +14 z^4 a^{-6} +20 z^4 a^{-8} +3 z^4 a^{-10} -6 z^4 a^{-12} +6 z^3 a^{-3} +12 z^3 a^{-5} +13 z^3 a^{-7} +14 z^3 a^{-9} +6 z^3 a^{-11} -z^3 a^{-13} +3 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2 a^{-6} -3 z^2 a^{-8} -z^2 a^{-10} +z^2 a^{-12} -z a^{-3} -4 z a^{-5} -3 z a^{-7} -z a^{-9} -z a^{-11} - a^{-2} - a^{-6} - a^{-8}
The A2 invariant Data:K11a117/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a117/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a152,}

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

Vassiliev invariants

V2 and V3: (3, 6)
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 48 72 222 34 576 1120 160 176 288 1152 2664 408 \frac{58191}{10} -\frac{1066}{15} \frac{35582}{15} \frac{593}{6} \frac{2991}{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=4 is the signature of K11a117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          3 3
19         51 -4
17        73  4
15       105   -5
13      97    2
11     910     1
9    79      -2
7   49       5
5  37        -4
3 15         4
1 2          -2
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\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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K11a116.gif

K11a116

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K11a118