K11a23

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K11a22.gif

K11a22

K11a24.gif

K11a24

Contents

K11a23.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a23 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X22,16,1,15 X20,18,21,17 X10,20,11,19 X16,22,17,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 12 2 18 14 6 22 20 10 16
A Braid Representative
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A Morse Link Presentation K11a23 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:K11a23/ThurstonBennequinNumber
Hyperbolic Volume 14.3488
A-Polynomial See Data:K11a23/A-polynomial

[edit Notes for K11a23'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 K11a23's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-10 t^2+24 t-31+24 t^{-1} -10 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 103, 2 }
Jones polynomial q^9-3 q^8+6 q^7-11 q^6+14 q^5-16 q^4+17 q^3-14 q^2+11 q-6+3 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+5 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -2 z^2+3 a^{-2} + a^{-4} -3 a^{-6} + a^{-8} -1
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +6 z^9 a^{-5} +3 z^9 a^{-7} +4 z^8 a^{-2} +7 z^8 a^{-4} +7 z^8 a^{-6} +4 z^8 a^{-8} +4 z^7 a^{-1} +z^7 a^{-3} -7 z^7 a^{-5} -z^7 a^{-7} +3 z^7 a^{-9} -z^6 a^{-2} -15 z^6 a^{-4} -21 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} +3 z^6+a z^5-4 z^5 a^{-1} -5 z^5 a^{-3} -4 z^5 a^{-5} -13 z^5 a^{-7} -9 z^5 a^{-9} -7 z^4 a^{-2} +12 z^4 a^{-4} +20 z^4 a^{-6} +4 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4-2 a z^3-2 z^3 a^{-1} +2 z^3 a^{-3} +11 z^3 a^{-5} +17 z^3 a^{-7} +8 z^3 a^{-9} +7 z^2 a^{-2} -4 z^2 a^{-4} -10 z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} +4 z^2+a z+2 z a^{-1} +z a^{-3} -5 z a^{-5} -8 z a^{-7} -3 z a^{-9} -3 a^{-2} + a^{-4} +3 a^{-6} + a^{-8} -1
The A2 invariant Data:K11a23/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a23/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_117, K11a111,}

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{28}{3} -\frac{4}{3} 64 -\frac{16}{3} \frac{32}{3} 8 \frac{256}{3} 32 \frac{224}{3} -\frac{32}{3} -\frac{1649}{15} -\frac{348}{5} \frac{4804}{45} -\frac{31}{9} \frac{271}{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=2 is the signature of K11a23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          2 -2
15         41 3
13        72  -5
11       74   3
9      97    -2
7     87     1
5    69      3
3   58       -3
1  27        5
-1 14         -3
-3 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a22.gif

K11a22

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K11a24