K11a175

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K11a174.gif

K11a174

K11a176.gif

K11a176

Contents

K11a175.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a175 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X16,7,17,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X10,20,11,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 12 14 16 18 2 22 6 20 10 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation K11a175 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a175/ThurstonBennequinNumber
Hyperbolic Volume 13.9605
A-Polynomial See Data:K11a175/A-polynomial

[edit Notes for K11a175's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a175's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+13 t^2-21 t+25-21 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 105, 0 }
Jones polynomial q^6-4 q^5+7 q^4-11 q^3+15 q^2-16 q+17-14 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +14 z^4-5 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +14 z^2-2 a^2-2 a^{-2} +5
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +4 a^2 z^8+10 z^8 a^{-2} +6 z^8 a^{-4} +8 z^8+4 a^3 z^7-12 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-3 a^2 z^6-35 z^6 a^{-2} -17 z^6 a^{-4} +z^6 a^{-6} -23 z^6+a^5 z^5-5 a^3 z^5-3 a z^5+3 z^5 a^{-1} -11 z^5 a^{-3} -11 z^5 a^{-5} -6 a^4 z^4+39 z^4 a^{-2} +13 z^4 a^{-4} -2 z^4 a^{-6} +30 z^4-2 a^5 z^3+2 a z^3+7 z^3 a^{-1} +13 z^3 a^{-3} +6 z^3 a^{-5} +3 a^4 z^2-3 a^2 z^2-17 z^2 a^{-2} -4 z^2 a^{-4} -19 z^2+a^5 z+a^3 z-a z-3 z a^{-1} -2 z a^{-3} +2 a^2+2 a^{-2} +5
The A2 invariant -q^{14}+q^{12}-2 q^{10}+q^8+q^6-2 q^4+4 q^2-2+3 q^{-2} + q^{-4} +3 q^{-8} -3 q^{-10} - q^{-14} - q^{-16} + q^{-18}
The G2 invariant q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}-2 q^{68}+16 q^{66}-28 q^{64}+40 q^{62}-44 q^{60}+31 q^{58}-7 q^{56}-33 q^{54}+73 q^{52}-105 q^{50}+116 q^{48}-99 q^{46}+48 q^{44}+28 q^{42}-114 q^{40}+187 q^{38}-217 q^{36}+188 q^{34}-106 q^{32}-21 q^{30}+149 q^{28}-237 q^{26}+260 q^{24}-186 q^{22}+57 q^{20}+82 q^{18}-179 q^{16}+186 q^{14}-106 q^{12}-28 q^{10}+153 q^8-206 q^6+154 q^4-3 q^2-182+332 q^{-2} -366 q^{-4} +266 q^{-6} -62 q^{-8} -177 q^{-10} +371 q^{-12} -440 q^{-14} +374 q^{-16} -184 q^{-18} -46 q^{-20} +244 q^{-22} -332 q^{-24} +289 q^{-26} -145 q^{-28} -33 q^{-30} +168 q^{-32} -213 q^{-34} +144 q^{-36} +5 q^{-38} -159 q^{-40} +256 q^{-42} -247 q^{-44} +122 q^{-46} +54 q^{-48} -226 q^{-50} +318 q^{-52} -304 q^{-54} +192 q^{-56} -27 q^{-58} -131 q^{-60} +228 q^{-62} -241 q^{-64} +183 q^{-66} -85 q^{-68} -13 q^{-70} +77 q^{-72} -103 q^{-74} +90 q^{-76} -55 q^{-78} +24 q^{-80} +5 q^{-82} -17 q^{-84} +17 q^{-86} -14 q^{-88} +7 q^{-90} -3 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a306,}

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

Vassiliev invariants

V2 and V3: (2, 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
8 0 32 \frac{76}{3} -\frac{28}{3} 0 32 32 0 \frac{256}{3} 0 \frac{608}{3} -\frac{224}{3} \frac{2551}{15} \frac{676}{15} -\frac{2996}{45} -\frac{7}{9} -\frac{89}{15}

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=0 is the signature of K11a175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         41 3
7        73  -4
5       84   4
3      87    -1
1     98     1
-1    69      3
-3   48       -4
-5  26        4
-7 14         -3
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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

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K11a174.gif

K11a174

K11a176.gif

K11a176