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

Visit K11a182 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X20,11,21,12 X2,13,3,14 X22,16,1,15 X8,17,9,18 X10,19,11,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -8
Dowker-Thistlethwaite code 4 12 14 16 18 20 2 22 8 10 6
A Braid Representative
A Morse Link Presentation K11a182 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:K11a182/ThurstonBennequinNumber
Hyperbolic Volume 11.2693
A-Polynomial See Data:K11a182/A-polynomial

[edit Notes for K11a182's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a182's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+11 t^2-13 t+13-13 t^{-1} +11 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 73, -4 }
Jones polynomial -q+3-4 q^{-1} +7 q^{-2} -9 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+3 z^2 a^8+a^8-2 z^6 a^6-9 z^4 a^6-11 z^2 a^6-4 a^6+z^8 a^4+6 z^6 a^4+13 z^4 a^4+13 z^2 a^4+4 a^4-z^6 a^2-4 z^4 a^2-3 z^2 a^2
Kauffman polynomial (db, data sources) z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-4 z^4 a^{10}+4 z^7 a^9-3 z^5 a^9-2 z^3 a^9+2 z a^9+4 z^8 a^8-7 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+3 z^9 a^7-6 z^7 a^7+5 z^5 a^7-2 z^3 a^7+z^{10} a^6+4 z^8 a^6-24 z^6 a^6+36 z^4 a^6-21 z^2 a^6+4 a^6+6 z^9 a^5-23 z^7 a^5+27 z^5 a^5-11 z^3 a^5+z^{10} a^4+3 z^8 a^4-27 z^6 a^4+42 z^4 a^4-23 z^2 a^4+4 a^4+3 z^9 a^3-12 z^7 a^3+12 z^5 a^3-4 z^3 a^3+z a^3+3 z^8 a^2-14 z^6 a^2+18 z^4 a^2-7 z^2 a^2+z^7 a-4 z^5 a+3 z^3 a
The A2 invariant q^{30}-q^{26}-2 q^{22}+q^{20}-q^{18}+q^{14}-2 q^{12}+3 q^{10}+2 q^6+q^4+1- q^{-2}
The G2 invariant q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+22 q^{144}-19 q^{142}+8 q^{140}+7 q^{138}-23 q^{136}+37 q^{134}-40 q^{132}+34 q^{130}-20 q^{128}-2 q^{126}+25 q^{124}-39 q^{122}+48 q^{120}-41 q^{118}+29 q^{116}-11 q^{114}-11 q^{112}+28 q^{110}-40 q^{108}+40 q^{106}-30 q^{104}+11 q^{102}+8 q^{100}-24 q^{98}+27 q^{96}-18 q^{94}-2 q^{92}+21 q^{90}-36 q^{88}+25 q^{86}+q^{84}-35 q^{82}+64 q^{80}-71 q^{78}+51 q^{76}-12 q^{74}-39 q^{72}+78 q^{70}-98 q^{68}+84 q^{66}-43 q^{64}-8 q^{62}+55 q^{60}-76 q^{58}+75 q^{56}-48 q^{54}+8 q^{52}+27 q^{50}-49 q^{48}+44 q^{46}-13 q^{44}-19 q^{42}+51 q^{40}-54 q^{38}+32 q^{36}+8 q^{34}-49 q^{32}+79 q^{30}-81 q^{28}+56 q^{26}-9 q^{24}-36 q^{22}+70 q^{20}-76 q^{18}+61 q^{16}-29 q^{14}-3 q^{12}+26 q^{10}-37 q^8+35 q^6-23 q^4+11 q^2+1-7 q^{-2} +6 q^{-4} -7 q^{-6} +4 q^{-8} -2 q^{-10} + q^{-12}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (2, -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
8 -16 32 \frac{124}{3} \frac{20}{3} -128 -\frac{448}{3} -\frac{256}{3} 16 \frac{256}{3} 128 \frac{992}{3} \frac{160}{3} \frac{8431}{15} \frac{3236}{15} \frac{10444}{45} -\frac{703}{9} \frac{511}{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=-4 is the signature of K11a182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          2 2
-1         21 -1
-3        52  3
-5       53   -2
-7      64    2
-9     55     0
-11    56      -1
-13   35       2
-15  25        -3
-17 13         2
-19 2          -2
-211           1
Integral Khovanov Homology

(db, data source)

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