K11a100

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K11a99.gif

K11a99

K11a101.gif

K11a101

Contents

K11a100.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a100 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X22,14,1,13 X8,16,9,15 X14,18,15,17 X6,20,7,19 X18,22,19,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -9, 8, -5, 9, -11, 10, -4, 11, -7
Dowker-Thistlethwaite code 4 10 12 20 16 2 22 8 14 6 18
A Braid Representative
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A Morse Link Presentation K11a100 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a100/ThurstonBennequinNumber
Hyperbolic Volume 16.1755
A-Polynomial See Data:K11a100/A-polynomial

[edit Notes for K11a100'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 K11a100's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+15 t^2-32 t+41-32 t^{-1} +15 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 141, 4 }
Jones polynomial -q^{11}+5 q^{10}-10 q^9+15 q^8-21 q^7+23 q^6-22 q^5+19 q^4-13 q^3+8 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -z^4 a^{-4} -6 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +2 z^2 a^{-4} -7 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} + a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +4 z^9 a^{-5} +9 z^9 a^{-7} +5 z^9 a^{-9} +5 z^8 a^{-4} +16 z^8 a^{-6} +21 z^8 a^{-8} +10 z^8 a^{-10} +3 z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} +14 z^7 a^{-9} +10 z^7 a^{-11} +z^6 a^{-2} -10 z^6 a^{-4} -41 z^6 a^{-6} -44 z^6 a^{-8} -9 z^6 a^{-10} +5 z^6 a^{-12} -7 z^5 a^{-3} -17 z^5 a^{-5} -38 z^5 a^{-7} -44 z^5 a^{-9} -15 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +7 z^4 a^{-4} +36 z^4 a^{-6} +27 z^4 a^{-8} -4 z^4 a^{-10} -5 z^4 a^{-12} +5 z^3 a^{-3} +18 z^3 a^{-5} +37 z^3 a^{-7} +30 z^3 a^{-9} +6 z^3 a^{-11} +3 z^2 a^{-2} -4 z^2 a^{-4} -15 z^2 a^{-6} -6 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -7 z a^{-5} -10 z a^{-7} -4 z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8}
The A2 invariant Data:K11a100/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a100/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a290,}

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

Vassiliev invariants

V2 and V3: (1, 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
4 0 8 \frac{14}{3} \frac{82}{3} 0 64 64 96 \frac{32}{3} 0 \frac{56}{3} \frac{328}{3} \frac{9151}{30} -\frac{2542}{15} \frac{30782}{45} \frac{1313}{18} \frac{1951}{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=4 is the signature of K11a100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          4 4
19         61 -5
17        94  5
15       126   -6
13      119    2
11     1112     1
9    811      -3
7   511       6
5  38        -5
3 16         5
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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a99.gif

K11a99

K11a101.gif

K11a101