K11n31

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K11n30.gif

K11n30

K11n32.gif

K11n32

Contents

K11n31.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n31 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,6,13,5 X2837 X9,17,10,16 X6,12,7,11 X13,20,14,21 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,19,22,18
Gauss code 1, -4, 2, -1, 3, -6, 4, -2, -5, 8, 6, -3, -7, 10, -8, 5, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 12 2 -16 6 -20 -10 -22 -14 -18
A Braid Representative
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A Morse Link Presentation K11n31 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n31/ThurstonBennequinNumber
Hyperbolic Volume 10.5314
A-Polynomial See Data:K11n31/A-polynomial

[edit Notes for K11n31'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 K11n31's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+2 t^2+2 t-5+2 t^{-1} +2 t^{-2} - t^{-3}
Conway polynomial -z^6-4 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 3, 2 }
Jones polynomial -q^{10}+2 q^9-2 q^8+2 q^7-q^6-q^3+2 q^2-q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +4 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} +3 a^{-2} -3 a^{-4} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-7} +z^9 a^{-9} +z^8 a^{-6} +3 z^8 a^{-8} +2 z^8 a^{-10} +z^7 a^{-3} -6 z^7 a^{-7} -4 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-2} +2 z^6 a^{-4} -7 z^6 a^{-6} -19 z^6 a^{-8} -11 z^6 a^{-10} -4 z^5 a^{-3} +8 z^5 a^{-7} -z^5 a^{-9} -5 z^5 a^{-11} -5 z^4 a^{-2} -10 z^4 a^{-4} +11 z^4 a^{-6} +32 z^4 a^{-8} +16 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} -4 z^3 a^{-7} +8 z^3 a^{-9} +6 z^3 a^{-11} +7 z^2 a^{-2} +10 z^2 a^{-4} -6 z^2 a^{-6} -17 z^2 a^{-8} -8 z^2 a^{-10} +z a^{-3} +3 z a^{-5} +2 z a^{-7} -2 z a^{-9} -2 z a^{-11} -3 a^{-2} -3 a^{-4} +2 a^{-8} + a^{-10}
The A2 invariant Data:K11n31/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n31/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: (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{446}{3} \frac{106}{3} 96 656 128 120 \frac{32}{3} 288 \frac{1784}{3} \frac{424}{3} \frac{79951}{30} -\frac{34}{5} \frac{54422}{45} \frac{977}{18} \frac{4111}{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 K11n31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
21           1-1
19          1 1
17         11 0
15       121  0
13      111   1
11     122    -1
9    121     0
7   111      -1
5  111       1
3 12         1
1            0
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11n30.gif

K11n30

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K11n32