K11a176

From Knot Atlas
Jump to: navigation, search

K11a175.gif

K11a175

K11a177.gif

K11a177

Contents

K11a176.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a176 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X16,8,17,7 X18,10,19,9 X2,12,3,11 X22,13,1,14 X20,15,21,16 X8,18,9,17 X6,20,7,19 X10,21,11,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 12 14 16 18 2 22 20 8 6 10
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11a176 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:K11a176/ThurstonBennequinNumber
Hyperbolic Volume 14.1348
A-Polynomial See Data:K11a176/A-polynomial

[edit Notes for K11a176's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a176's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-13 t^2+23 t-27+23 t^{-1} -13 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-3 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 111, 2 }
Jones polynomial -q^8+4 q^7-8 q^6+12 q^5-16 q^4+18 q^3-17 q^2+15 q-10+6 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-14 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +4 z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +8 z^8 a^{-2} +11 z^8 a^{-4} +7 z^8 a^{-6} +4 z^8+3 a z^7-2 z^7 a^{-1} -10 z^7 a^{-3} +2 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-26 z^6 a^{-2} -29 z^6 a^{-4} -10 z^6 a^{-6} +4 z^6 a^{-8} -10 z^6-9 a z^5-7 z^5 a^{-1} +z^5 a^{-3} -14 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+32 z^4 a^{-2} +31 z^4 a^{-4} +4 z^4 a^{-6} -6 z^4 a^{-8} +8 z^4+7 a z^3+7 z^3 a^{-1} +6 z^3 a^{-3} +12 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +2 a^2 z^2-19 z^2 a^{-2} -15 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} -5 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2
The A2 invariant q^8-q^6+2 q^4-q^2-1+3 q^{-2} -3 q^{-4} +4 q^{-6} - q^{-8} + q^{-12} -3 q^{-14} +3 q^{-16} - q^{-18} + q^{-22} - q^{-24}
The G2 invariant q^{46}-2 q^{44}+5 q^{42}-9 q^{40}+10 q^{38}-10 q^{36}+2 q^{34}+15 q^{32}-33 q^{30}+54 q^{28}-64 q^{26}+54 q^{24}-22 q^{22}-37 q^{20}+107 q^{18}-165 q^{16}+190 q^{14}-155 q^{12}+60 q^{10}+81 q^8-220 q^6+320 q^4-326 q^2+227-48 q^{-2} -161 q^{-4} +313 q^{-6} -357 q^{-8} +279 q^{-10} -98 q^{-12} -98 q^{-14} +232 q^{-16} -254 q^{-18} +142 q^{-20} +45 q^{-22} -225 q^{-24} +306 q^{-26} -245 q^{-28} +52 q^{-30} +202 q^{-32} -413 q^{-34} +503 q^{-36} -419 q^{-38} +184 q^{-40} +124 q^{-42} -401 q^{-44} +544 q^{-46} -504 q^{-48} +309 q^{-50} -29 q^{-52} -222 q^{-54} +359 q^{-56} -340 q^{-58} +191 q^{-60} +17 q^{-62} -189 q^{-64} +243 q^{-66} -167 q^{-68} -6 q^{-70} +197 q^{-72} -313 q^{-74} +315 q^{-76} -192 q^{-78} -7 q^{-80} +203 q^{-82} -339 q^{-84} +362 q^{-86} -274 q^{-88} +118 q^{-90} +50 q^{-92} -179 q^{-94} +237 q^{-96} -219 q^{-98} +152 q^{-100} -60 q^{-102} -23 q^{-104} +72 q^{-106} -92 q^{-108} +77 q^{-110} -48 q^{-112} +22 q^{-114} +2 q^{-116} -13 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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 48 24 0 \frac{272}{3} -\frac{64}{3} 72 0 32 0 0 168 -24 40 72 -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=2 is the signature of K11a176. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         51 -4
11        73  4
9       95   -4
7      97    2
5     89     1
3    79      -2
1   49       5
-1  26        -4
-3 14         3
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

K11a175.gif

K11a175

K11a177.gif

K11a177