K11n176

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K11n175.gif

K11n175

K11n177.gif

K11n177

Contents

K11n176.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n176 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X5,16,6,17 X18,7,19,8 X20,10,21,9 X2,11,3,12 X8,13,9,14 X15,4,16,5 X22,17,1,18 X12,20,13,19 X14,21,15,22
Gauss code 1, -6, 2, 8, -3, -1, 4, -7, 5, -2, 6, -10, 7, -11, -8, 3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 -16 18 20 2 8 -4 22 12 14
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n176 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n176/ThurstonBennequinNumber
Hyperbolic Volume 14.5345
A-Polynomial See Data:K11n176/A-polynomial

[edit Notes for K11n176's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n176's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+15 t-19+15 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, -2 }
Jones polynomial -1+5 q^{-1} -7 q^{-2} +10 q^{-3} -11 q^{-4} +10 q^{-5} -9 q^{-6} +6 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-4 z^2 a^6-2 a^6+z^6 a^4+3 z^4 a^4+3 z^2 a^4-z^4 a^2+2 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-9 z^5 a^9+7 z^3 a^9-2 z a^9+4 z^8 a^8-11 z^6 a^8+7 z^4 a^8-3 z^2 a^8+a^8+2 z^9 a^7-11 z^5 a^7+8 z^3 a^7-2 z a^7+8 z^8 a^6-23 z^6 a^6+23 z^4 a^6-12 z^2 a^6+2 a^6+2 z^9 a^5-z^7 a^5-2 z^5 a^5+2 z a^5+4 z^8 a^4-11 z^6 a^4+18 z^4 a^4-9 z^2 a^4+2 z^7 a^3+2 z a^3+5 z^4 a^2-2 z^2 a^2-2 a^2+z^3 a
The A2 invariant q^{28}-q^{24}+2 q^{22}-2 q^{20}-3 q^{14}+q^{12}-2 q^{10}+3 q^8+2 q^6+3 q^2-1
The G2 invariant Data:K11n176/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n125,}

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

Vassiliev invariants

V2 and V3: (0, 2)
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
0 16 0 -64 0 0 \frac{352}{3} -\frac{32}{3} -16 0 128 0 0 160 -\frac{448}{3} 384 32 64

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 K11n176. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-101χ
1         1-1
-1        4 4
-3       42 -2
-5      63  3
-7     54   -1
-9    56    -1
-11   45     1
-13  25      -3
-15 14       3
-17 2        -2
-191         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\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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K11n175.gif

K11n175

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K11n177