K11a102

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K11a101.gif

K11a101

K11a103.gif

K11a103

Contents

K11a102.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a102 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X18,9,19,10 X2,11,3,12 X6,14,7,13 X20,16,21,15 X22,18,1,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -4, 7, -3, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 14 12 18 2 6 20 22 8 16
A Braid Representative
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A Morse Link Presentation K11a102 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:K11a102/ThurstonBennequinNumber
Hyperbolic Volume 14.6861
A-Polynomial See Data:K11a102/A-polynomial

[edit Notes for K11a102's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant -2

[edit Notes for K11a102's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+23 t-27+23 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 99, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+11 q^5-14 q^4+16 q^3-16 q^2+13 q-10+6 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-3 z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} -4 z^2+2 a^2-3 a^{-2} +4 a^{-4} - a^{-6} -1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +6 z^8 a^{-2} +9 z^8 a^{-4} +6 z^8 a^{-6} +3 z^8+2 a z^7-6 z^7 a^{-1} -18 z^7 a^{-3} -5 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-22 z^6 a^{-2} -32 z^6 a^{-4} -14 z^6 a^{-6} +3 z^6 a^{-8} -6 z^6-5 a z^5+9 z^5 a^{-1} +27 z^5 a^{-3} +2 z^5 a^{-5} -10 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+31 z^4 a^{-2} +48 z^4 a^{-4} +16 z^4 a^{-6} -6 z^4 a^{-8} +z^4+2 a z^3-16 z^3 a^{-1} -24 z^3 a^{-3} +2 z^3 a^{-5} +6 z^3 a^{-7} -2 z^3 a^{-9} +5 a^2 z^2-22 z^2 a^{-2} -28 z^2 a^{-4} -7 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2+a z+8 z a^{-1} +9 z a^{-3} +z a^{-5} -z a^{-7} -2 a^2+3 a^{-2} +4 a^{-4} + a^{-6} -1
The A2 invariant Data:K11a102/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a102/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a181, K11a199,}

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

Vassiliev invariants

V2 and V3: (-3, 0)
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 0 72 114 30 0 -32 -64 0 -288 0 -1368 -360 -\frac{15551}{10} -\frac{1458}{5} -\frac{9742}{15} \frac{383}{6} -\frac{831}{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=2 is the signature of K11a102. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        72  5
9       74   -3
7      97    2
5     77     0
3    69      -3
1   58       3
-1  15        -4
-3 15         4
-5 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a101.gif

K11a101

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K11a103