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

Visit K11a62 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,15,1,16 X6,17,7,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 16 10 2 18 20 22 6 12 14
A Braid Representative
A Morse Link Presentation K11a62 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a62/ThurstonBennequinNumber
Hyperbolic Volume 10.6386
A-Polynomial See Data:K11a62/A-polynomial

[edit Notes for K11a62's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 6

[edit Notes for K11a62's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-8 t^2+9 t-9+9 t^{-1} -8 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6+2 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 55, -6 }
Jones polynomial  q^{-1} -2 q^{-2} +4 q^{-3} -5 q^{-4} +7 q^{-5} -8 q^{-6} +8 q^{-7} -7 q^{-8} +6 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{10}-4 z^2 a^{10}-3 a^{10}+2 z^6 a^8+10 z^4 a^8+14 z^2 a^8+6 a^8-z^8 a^6-6 z^6 a^6-12 z^4 a^6-11 z^2 a^6-5 a^6+z^6 a^4+5 z^4 a^4+7 z^2 a^4+3 a^4
Kauffman polynomial (db, data sources) z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-2 z^3 a^{13}+z a^{13}+4 z^6 a^{12}-6 z^4 a^{12}+4 z^2 a^{12}+4 z^7 a^{11}-7 z^5 a^{11}+2 z^3 a^{11}+4 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-9 z^2 a^{10}+3 a^{10}+3 z^9 a^9-9 z^7 a^9+6 z^5 a^9-3 z^3 a^9+z^{10} a^8+2 z^8 a^8-24 z^6 a^8+41 z^4 a^8-27 z^2 a^8+6 a^8+5 z^9 a^7-24 z^7 a^7+34 z^5 a^7-17 z^3 a^7+3 z a^7+z^{10} a^6-z^8 a^6-15 z^6 a^6+35 z^4 a^6-23 z^2 a^6+5 a^6+2 z^9 a^5-11 z^7 a^5+18 z^5 a^5-9 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+12 z^4 a^4-10 z^2 a^4+3 a^4
The A2 invariant -q^{36}-q^{34}-q^{30}+q^{28}+2 q^{24}+q^{22}-q^{20}+q^{18}-2 q^{16}+q^{14}+q^{10}+q^8+q^4
The G2 invariant q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+4 q^{182}-5 q^{180}+6 q^{178}-6 q^{176}+3 q^{174}+2 q^{172}-6 q^{170}+11 q^{168}-12 q^{166}+10 q^{164}-8 q^{162}-q^{160}+7 q^{158}-14 q^{156}+15 q^{154}-12 q^{152}+5 q^{150}-6 q^{146}+8 q^{144}-8 q^{142}+6 q^{140}-6 q^{138}+4 q^{136}-5 q^{134}+3 q^{132}+2 q^{130}-8 q^{128}+14 q^{126}-13 q^{124}+7 q^{122}+q^{120}-10 q^{118}+11 q^{116}-4 q^{114}-5 q^{112}+14 q^{110}-14 q^{108}+7 q^{106}+11 q^{104}-24 q^{102}+31 q^{100}-26 q^{98}+12 q^{96}+7 q^{94}-21 q^{92}+33 q^{90}-30 q^{88}+23 q^{86}-9 q^{84}-7 q^{82}+19 q^{80}-24 q^{78}+18 q^{76}-9 q^{74}-6 q^{72}+16 q^{70}-18 q^{68}+7 q^{66}+10 q^{64}-25 q^{62}+29 q^{60}-22 q^{58}+q^{56}+20 q^{54}-34 q^{52}+38 q^{50}-26 q^{48}+9 q^{46}+11 q^{44}-22 q^{42}+25 q^{40}-17 q^{38}+10 q^{36}-4 q^{32}+6 q^{30}-5 q^{28}+4 q^{26}-q^{24}+q^{22}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (6, -17)
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
24 -136 288 876 124 -3264 -\frac{18832}{3} -\frac{3232}{3} -776 2304 9248 21024 2976 \frac{231671}{5} \frac{28148}{15} \frac{84188}{5} 339 \frac{10391}{5}

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=-6 is the signature of K11a62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1           11
-3          1 -1
-5         31 2
-7        32  -1
-9       42   2
-11      43    -1
-13     44     0
-15    34      1
-17   34       -1
-19  13        2
-21 13         -2
-23 1          1
-251           -1
Integral Khovanov Homology

(db, data source)

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

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