K11a324

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K11a323.gif

K11a323

K11a325.gif

K11a325

Contents

K11a324.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a324 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X16,5,17,6 X22,8,1,7 X20,9,21,10 X18,11,19,12 X2,13,3,14 X8,15,9,16 X4,17,5,18 X10,19,11,20 X14,21,15,22
Gauss code 1, -7, 2, -9, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 20 18 2 8 4 10 14
A Braid Representative
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A Morse Link Presentation K11a324 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a324/ThurstonBennequinNumber
Hyperbolic Volume 15.0339
A-Polynomial See Data:K11a324/A-polynomial

[edit Notes for K11a324's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a324's four dimensional invariants]

Polynomial invariants

Alexander polynomial -5 t^2+25 t-39+25 t^{-1} -5 t^{-2}
Conway polynomial -5 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 99, -2 }
Jones polynomial q-3+6 q^{-1} -10 q^{-2} +14 q^{-3} -15 q^{-4} +16 q^{-5} -13 q^{-6} +10 q^{-7} -7 q^{-8} +3 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+3 z^2 a^8+a^8-2 z^4 a^6-z^2 a^6-a^6-2 z^4 a^4+z^2 a^4+2 a^4-z^4 a^2+z^2 a^2+z^2
Kauffman polynomial (db, data sources) z^7 a^{11}-4 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-11 z^6 a^{10}+11 z^4 a^{10}-4 z^2 a^{10}+a^{10}+4 z^9 a^9-13 z^7 a^9+10 z^5 a^9-4 z^3 a^9+3 z a^9+2 z^{10} a^8+z^8 a^8-22 z^6 a^8+28 z^4 a^8-12 z^2 a^8+a^8+10 z^9 a^7-34 z^7 a^7+38 z^5 a^7-23 z^3 a^7+7 z a^7+2 z^{10} a^6+6 z^8 a^6-35 z^6 a^6+45 z^4 a^6-21 z^2 a^6+a^6+6 z^9 a^5-13 z^7 a^5+10 z^5 a^5-4 z^3 a^5+2 z a^5+8 z^8 a^4-19 z^6 a^4+22 z^4 a^4-11 z^2 a^4+2 a^4+7 z^7 a^3-11 z^5 a^3+7 z^3 a^3+5 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-3 z^3 a+z^4-z^2
The A2 invariant Data:K11a324/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a324/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: (5, -12)
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
20 -96 200 \frac{1846}{3} \frac{410}{3} -1920 -4096 -672 -896 \frac{4000}{3} 4608 \frac{36920}{3} \frac{8200}{3} \frac{162607}{6} -2778 \frac{139790}{9} \frac{7573}{18} \frac{14767}{6}

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 K11a324. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
3           11
1          2 -2
-1         41 3
-3        73  -4
-5       73   4
-7      87    -1
-9     87     1
-11    58      3
-13   58       -3
-15  25        3
-17 15         -4
-19 2          2
-211           -1
Integral Khovanov Homology

(db, data source)

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