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

Visit K11a7 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X6,16,7,15 X22,17,1,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 16 2 18 20 6 22 12 14
A Braid Representative
A Morse Link Presentation K11a7 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a7/ThurstonBennequinNumber
Hyperbolic Volume 13.5338
A-Polynomial See Data:K11a7/A-polynomial

[edit Notes for K11a7'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 K11a7's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 95, -2 }
Jones polynomial q^3-3 q^2+6 q-9+13 q^{-1} -15 q^{-2} +15 q^{-3} -13 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+13 a^4 z^2-15 a^2 z^2+5 z^2-2 a^6+6 a^4-6 a^2+3
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+7 a^3 z^9+4 a z^9+4 a^6 z^8+7 a^4 z^8+8 a^2 z^8+5 z^8+4 a^7 z^7-2 a^5 z^7-19 a^3 z^7-10 a z^7+3 z^7 a^{-1} +3 a^8 z^6-4 a^6 z^6-26 a^4 z^6-37 a^2 z^6+z^6 a^{-2} -17 z^6+a^9 z^5-5 a^7 z^5+a^5 z^5+21 a^3 z^5+5 a z^5-9 z^5 a^{-1} -6 a^8 z^4+a^6 z^4+41 a^4 z^4+55 a^2 z^4-3 z^4 a^{-2} +18 z^4-2 a^9 z^3-a^7 z^3-2 a^5 z^3-6 a^3 z^3+2 a z^3+5 z^3 a^{-1} +3 a^8 z^2-4 a^6 z^2-27 a^4 z^2-31 a^2 z^2+z^2 a^{-2} -10 z^2+a^9 z+a^7 z-a z-z a^{-1} +2 a^6+6 a^4+6 a^2+3
The A2 invariant -q^{24}-q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-2 q^8+2 q^6-4 q^4+2 q^2+2 q^{-4} - q^{-6} + q^{-8}
The G2 invariant q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-6 q^{118}-2 q^{116}+16 q^{114}-29 q^{112}+41 q^{110}-42 q^{108}+27 q^{106}-2 q^{104}-37 q^{102}+75 q^{100}-101 q^{98}+101 q^{96}-76 q^{94}+20 q^{92}+47 q^{90}-112 q^{88}+160 q^{86}-165 q^{84}+125 q^{82}-45 q^{80}-57 q^{78}+139 q^{76}-178 q^{74}+161 q^{72}-85 q^{70}-16 q^{68}+105 q^{66}-137 q^{64}+101 q^{62}-3 q^{60}-103 q^{58}+170 q^{56}-158 q^{54}+63 q^{52}+80 q^{50}-214 q^{48}+290 q^{46}-259 q^{44}+139 q^{42}+36 q^{40}-202 q^{38}+298 q^{36}-296 q^{34}+199 q^{32}-49 q^{30}-102 q^{28}+199 q^{26}-211 q^{24}+142 q^{22}-22 q^{20}-97 q^{18}+151 q^{16}-137 q^{14}+42 q^{12}+80 q^{10}-175 q^8+207 q^6-153 q^4+34 q^2+104-206 q^{-2} +234 q^{-4} -184 q^{-6} +82 q^{-8} +34 q^{-10} -121 q^{-12} +160 q^{-14} -140 q^{-16} +92 q^{-18} -25 q^{-20} -26 q^{-22} +52 q^{-24} -58 q^{-26} +45 q^{-28} -26 q^{-30} +10 q^{-32} +4 q^{-34} -9 q^{-36} +8 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a33, K11a82,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a325,}

Vassiliev invariants

V2 and V3: (0, -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
0 -16 0 64 16 0 -\frac{160}{3} \frac{32}{3} 16 0 128 0 0 128 \frac{560}{3} -\frac{976}{3} \frac{128}{3} -64

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 K11a7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          2 -2
3         41 3
1        52  -3
-1       84   4
-3      86    -2
-5     77     0
-7    68      2
-9   47       -3
-11  26        4
-13 14         -3
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

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