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

Visit K11a206 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,5,17,6 X18,7,19,8 X20,9,21,10 X14,12,15,11 X2,13,3,14 X22,15,1,16 X6,17,7,18 X8,19,9,20 X10,21,11,22
Gauss code 1, -7, 2, -1, 3, -9, 4, -10, 5, -11, 6, -2, 7, -6, 8, -3, 9, -4, 10, -5, 11, -8
Dowker-Thistlethwaite code 4 12 16 18 20 14 2 22 6 8 10
A Braid Representative
A Morse Link Presentation K11a206 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a206/ThurstonBennequinNumber
Hyperbolic Volume 8.62876
A-Polynomial See Data:K11a206/A-polynomial

[edit Notes for K11a206'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 K11a206's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-7 t^2+7 t-7+7 t^{-1} -7 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6+3 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 47, -6 }
Jones polynomial  q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +6 q^{-5} -6 q^{-6} +7 q^{-7} -6 q^{-8} +5 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+13 z^2 a^8+4 a^8-z^8 a^6-6 z^6 a^6-11 z^4 a^6-7 z^2 a^6-a^6+z^6 a^4+5 z^4 a^4+6 z^2 a^4+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}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-3 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+3 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-11 z^2 a^{10}+3 a^{10}+2 z^9 a^9-5 z^7 a^9-z^5 a^9+5 z^3 a^9-3 z a^9+z^{10} a^8-z^8 a^8-10 z^6 a^8+19 z^4 a^8-13 z^2 a^8+4 a^8+4 z^9 a^7-20 z^7 a^7+29 z^5 a^7-14 z^3 a^7+2 z a^7+z^{10} a^6-3 z^8 a^6-4 z^6 a^6+14 z^4 a^6-7 z^2 a^6+a^6+2 z^9 a^5-12 z^7 a^5+22 z^5 a^5-13 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+11 z^4 a^4-7 z^2 a^4+a^4
The A2 invariant -q^{36}-q^{34}-q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+3 q^{18}+q^{14}-q^{12}+q^4
The G2 invariant q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+5 q^{182}-6 q^{180}+6 q^{178}-5 q^{176}+q^{174}+3 q^{172}-7 q^{170}+11 q^{168}-10 q^{166}+7 q^{164}-3 q^{162}-3 q^{160}+6 q^{158}-9 q^{156}+9 q^{154}-8 q^{152}+3 q^{150}+q^{148}-6 q^{146}+6 q^{144}-5 q^{142}+2 q^{140}-2 q^{138}-q^{136}-q^{134}-q^{130}+q^{126}-4 q^{124}+3 q^{122}-4 q^{120}+q^{118}-3 q^{112}+6 q^{110}-4 q^{108}-2 q^{106}+9 q^{104}-14 q^{102}+16 q^{100}-9 q^{98}+3 q^{96}+8 q^{94}-11 q^{92}+19 q^{90}-13 q^{88}+9 q^{86}-q^{84}-3 q^{82}+9 q^{80}-7 q^{78}+7 q^{76}-q^{74}-2 q^{72}+4 q^{70}-5 q^{68}-q^{66}+8 q^{64}-14 q^{62}+13 q^{60}-8 q^{58}-2 q^{56}+13 q^{54}-20 q^{52}+19 q^{50}-14 q^{48}+4 q^{46}+6 q^{44}-13 q^{42}+15 q^{40}-10 q^{38}+7 q^{36}-2 q^{32}+3 q^{30}-4 q^{28}+3 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: (8, -23)
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
32 -184 512 \frac{4048}{3} \frac{680}{3} -5888 -\frac{32656}{3} -\frac{5728}{3} -1624 \frac{16384}{3} 16928 \frac{129536}{3} \frac{21760}{3} \frac{1334044}{15} \frac{6184}{15} \frac{1694776}{45} \frac{6308}{9} \frac{77164}{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=-6 is the signature of K11a206. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1           11
-3          1 -1
-5         21 1
-7        32  -1
-9       31   2
-11      33    0
-13     43     1
-15    23      1
-17   34       -1
-19  12        1
-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}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
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.

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

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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