K11a233

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K11a232.gif

K11a232

K11a234.gif

K11a234

Contents

K11a233.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a233 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X18,11,19,12 X2,14,3,13 X8,15,9,16 X6,18,7,17 X12,19,13,20 X16,22,17,21
Gauss code 1, -7, 2, -1, 3, -9, 4, -8, 5, -3, 6, -10, 7, -2, 8, -11, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 14 10 20 22 18 2 8 6 12 16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a233 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a233/ThurstonBennequinNumber
Hyperbolic Volume 18.7296
A-Polynomial See Data:K11a233/A-polynomial

[edit Notes for K11a233'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 K11a233's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+19 t^2-37 t+47-37 t^{-1} +19 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 173, 0 }
Jones polynomial q^6-5 q^5+11 q^4-18 q^3+25 q^2-28 q+28-24 q^{-1} +18 q^{-2} -10 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^4-6 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} +10 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10}+10 a z^9+19 z^9 a^{-1} +9 z^9 a^{-3} +13 a^2 z^8+21 z^8 a^{-2} +10 z^8 a^{-4} +24 z^8+9 a^3 z^7-4 a z^7-26 z^7 a^{-1} -8 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-21 a^2 z^6-62 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -65 z^6+a^5 z^5-12 a^3 z^5-16 a z^5-8 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+18 a^2 z^4+48 z^4 a^{-2} +13 z^4 a^{-4} -z^4 a^{-6} +56 z^4-a^5 z^3+8 a^3 z^3+17 a z^3+17 z^3 a^{-1} +13 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-14 z^2 a^{-2} -2 z^2 a^{-4} -21 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3
The A2 invariant -q^{14}+2 q^{12}-4 q^{10}+3 q^8+2 q^6-4 q^4+6 q^2-5+4 q^{-2} - q^{-6} +5 q^{-8} -5 q^{-10} +2 q^{-12} -2 q^{-16} + q^{-18}
The G2 invariant q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+17 q^{72}-19 q^{70}+12 q^{68}+10 q^{66}-43 q^{64}+89 q^{62}-134 q^{60}+156 q^{58}-137 q^{56}+47 q^{54}+124 q^{52}-341 q^{50}+547 q^{48}-650 q^{46}+539 q^{44}-191 q^{42}-364 q^{40}+970 q^{38}-1370 q^{36}+1362 q^{34}-847 q^{32}-58 q^{30}+1029 q^{28}-1692 q^{26}+1741 q^{24}-1115 q^{22}+53 q^{20}+995 q^{18}-1553 q^{16}+1370 q^{14}-502 q^{12}-631 q^{10}+1504 q^8-1703 q^6+1078 q^4+119 q^2-1412+2279 q^{-2} -2327 q^{-4} +1526 q^{-6} -137 q^{-8} -1329 q^{-10} +2335 q^{-12} -2541 q^{-14} +1875 q^{-16} -610 q^{-18} -769 q^{-20} +1750 q^{-22} -1955 q^{-24} +1364 q^{-26} -237 q^{-28} -899 q^{-30} +1524 q^{-32} -1392 q^{-34} +552 q^{-36} +590 q^{-38} -1509 q^{-40} +1818 q^{-42} -1366 q^{-44} +367 q^{-46} +751 q^{-48} -1552 q^{-50} +1747 q^{-52} -1340 q^{-54} +554 q^{-56} +263 q^{-58} -838 q^{-60} +1029 q^{-62} -862 q^{-64} +498 q^{-66} -99 q^{-68} -190 q^{-70} +312 q^{-72} -306 q^{-74} +213 q^{-76} -101 q^{-78} +20 q^{-80} +30 q^{-82} -43 q^{-84} +36 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (1, 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
4 0 8 -\frac{34}{3} -\frac{62}{3} 0 0 32 -32 \frac{32}{3} 0 -\frac{136}{3} -\frac{248}{3} -\frac{1649}{30} \frac{86}{5} -\frac{3778}{45} -\frac{271}{18} -\frac{689}{30}

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 K11a233. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          4 -4
9         71 6
7        114  -7
5       147   7
3      1411    -3
1     1414     0
-1    1115      4
-3   713       -6
-5  311        8
-7 17         -6
-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}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a232.gif

K11a232

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K11a234