K11a160

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K11a159.gif

K11a159

K11a161.gif

K11a161

Contents

K11a160.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a160 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,5,19,6 X14,7,15,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X8,15,9,16 X12,18,13,17 X6,19,7,20 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 10 18 14 2 22 20 8 12 6 16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a160 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a160/ThurstonBennequinNumber
Hyperbolic Volume 17.0985
A-Polynomial See Data:K11a160/A-polynomial

[edit Notes for K11a160'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 K11a160's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+17 t^2-30 t+37-30 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 145, 0 }
Jones polynomial q^6-4 q^5+9 q^4-15 q^3+20 q^2-23 q+24-20 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-8 z^2 a^{-2} +2 z^2 a^{-4} +9 z^2-a^2-3 a^{-2} + a^{-4} +4
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+13 z^9 a^{-1} +6 z^9 a^{-3} +10 a^2 z^8+15 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+8 a^3 z^7-a z^7-17 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-16 a^2 z^6-47 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -51 z^6+a^5 z^5-12 a^3 z^5-17 a z^5-11 z^5 a^{-1} -16 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+12 a^2 z^4+44 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+24 z^3 a^{-1} +18 z^3 a^{-3} +6 z^3 a^{-5} +a^4 z^2-6 a^2 z^2-18 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -20 z^2-2 a^3 z-6 a z-8 z a^{-1} -6 z a^{-3} -2 z a^{-5} +a^2+3 a^{-2} + a^{-4} +4
The A2 invariant -q^{14}+2 q^{12}-3 q^{10}+2 q^8+q^6-3 q^4+6 q^2-3+4 q^{-2} - q^{-4} -2 q^{-6} +3 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-16} + q^{-18}
The G2 invariant q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-16 q^{70}+7 q^{68}+16 q^{66}-46 q^{64}+84 q^{62}-113 q^{60}+111 q^{58}-73 q^{56}-15 q^{54}+143 q^{52}-275 q^{50}+376 q^{48}-386 q^{46}+258 q^{44}-5 q^{42}-327 q^{40}+628 q^{38}-780 q^{36}+692 q^{34}-355 q^{32}-146 q^{30}+635 q^{28}-907 q^{26}+852 q^{24}-467 q^{22}-87 q^{20}+565 q^{18}-770 q^{16}+591 q^{14}-104 q^{12}-456 q^{10}+848 q^8-849 q^6+439 q^4+234 q^2-898+1280 q^{-2} -1218 q^{-4} +713 q^{-6} +58 q^{-8} -812 q^{-10} +1298 q^{-12} -1328 q^{-14} +916 q^{-16} -225 q^{-18} -485 q^{-20} +928 q^{-22} -967 q^{-24} +608 q^{-26} -22 q^{-28} -518 q^{-30} +780 q^{-32} -647 q^{-34} +179 q^{-36} +404 q^{-38} -847 q^{-40} +945 q^{-42} -667 q^{-44} +124 q^{-46} +452 q^{-48} -845 q^{-50} +935 q^{-52} -705 q^{-54} +288 q^{-56} +144 q^{-58} -455 q^{-60} +556 q^{-62} -473 q^{-64} +285 q^{-66} -69 q^{-68} -91 q^{-70} +170 q^{-72} -173 q^{-74} +127 q^{-76} -66 q^{-78} +18 q^{-80} +13 q^{-82} -25 q^{-84} +22 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a76, K11a289,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a76, K11a289,}

Vassiliev invariants

V2 and V3: (0, -1)
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 -8 0 -16 -8 0 \frac{112}{3} \frac{64}{3} 24 0 32 0 0 104 -104 \frac{632}{3} -\frac{88}{3} 40

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 K11a160. 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         61 5
7        93  -6
5       116   5
3      129    -3
1     1211     1
-1    913      4
-3   611       -5
-5  39        6
-7 16         -5
-9 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

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K11a159.gif

K11a159

K11a161.gif

K11a161