K11n108

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K11n107.gif

K11n107

K11n109.gif

K11n109

Contents

K11n108.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n108 at Knotilus!


Knot K11n108.
A graph, K11n108.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,16,8,17 X18,9,19,10 X2,11,3,12 X20,13,21,14 X15,6,16,7 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code 1, -6, 2, -1, -3, 8, -4, -11, 5, -2, 6, -10, 7, 3, -8, 4, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 -14 -16 18 2 20 -6 22 12 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11n108 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:K11n108/ThurstonBennequinNumber
Hyperbolic Volume 14.6102
A-Polynomial See Data:K11n108/A-polynomial

[edit Notes for K11n108'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 K11n108's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+8 t^2-17 t+21-17 t^{-1} +8 t^{-2} - t^{-3}
Conway polynomial -z^6+2 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 73, -4 }
Jones polynomial 2 q^{-2} -5 q^{-3} +9 q^{-4} -11 q^{-5} +13 q^{-6} -12 q^{-7} +10 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}+2 z^4 a^8+3 z^2 a^8-z^6 a^6-2 z^4 a^6+a^6+2 z^4 a^4+4 z^2 a^4+a^4
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+5 z^7 a^{11}-9 z^5 a^{11}+6 z^3 a^{11}-3 z a^{11}+4 z^8 a^{10}-3 z^6 a^{10}-2 z^4 a^{10}-z^2 a^{10}+a^{10}+z^9 a^9+9 z^7 a^9-20 z^5 a^9+13 z^3 a^9-3 z a^9+6 z^8 a^8-5 z^6 a^8-3 z^4 a^8+3 z^2 a^8+z^9 a^7+5 z^7 a^7-7 z^5 a^7+z^3 a^7+z a^7+2 z^8 a^6+z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+z^7 a^5+3 z^5 a^5-4 z^3 a^5+3 z^4 a^4-4 z^2 a^4+a^4
The A2 invariant -q^{34}+q^{30}-3 q^{28}+q^{26}-q^{24}-q^{22}+3 q^{20}-q^{18}+4 q^{16}-q^{14}+2 q^{10}-2 q^8+2 q^6
The G2 invariant Data:K11n108/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: (6, -15)
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
24 -120 288 764 124 -2880 -5360 -928 -792 2304 7200 18336 2976 \frac{190431}{5} \frac{636}{5} \frac{80108}{5} 363 \frac{10831}{5}

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 K11n108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-10χ
-3         22
-5        41-3
-7       51 4
-9      64  -2
-11     75   2
-13    56    1
-15   57     -2
-17  25      3
-19 15       -4
-21 2        2
-231         -1
Integral Khovanov Homology

(db, data source)

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

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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K11n107.gif

K11n107

K11n109.gif

K11n109