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

Visit K11a151 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,5,17,6 X20,7,21,8 X14,10,15,9 X2,11,3,12 X18,14,19,13 X22,15,1,16 X12,18,13,17 X6,19,7,20 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -5, 8, -3, 9, -7, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 10 16 20 14 2 18 22 12 6 8
A Braid Representative
A Morse Link Presentation K11a151 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a151/ThurstonBennequinNumber
Hyperbolic Volume 16.1582
A-Polynomial See Data:K11a151/A-polynomial

[edit Notes for K11a151'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 K11a151's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-15 t^2+26 t-31+26 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 127, -2 }
Jones polynomial q^3-4 q^2+8 q-13+18 q^{-1} -20 q^{-2} +21 q^{-3} -17 q^{-4} +13 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+8 a^4 z^4-9 a^2 z^4+3 z^4-3 a^6 z^2+11 a^4 z^2-6 a^2 z^2+2 z^2-3 a^6+5 a^4-a^2
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+11 a^3 z^9+6 a z^9+7 a^6 z^8+9 a^4 z^8+9 a^2 z^8+7 z^8+6 a^7 z^7-21 a^3 z^7-11 a z^7+4 z^7 a^{-1} +3 a^8 z^6-10 a^6 z^6-25 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -19 z^6+a^9 z^5-10 a^7 z^5-13 a^5 z^5+10 a^3 z^5+2 a z^5-10 z^5 a^{-1} -4 a^8 z^4+8 a^6 z^4+25 a^4 z^4+30 a^2 z^4-2 z^4 a^{-2} +15 z^4-2 a^9 z^3+8 a^7 z^3+17 a^5 z^3+5 a^3 z^3+3 a z^3+5 z^3 a^{-1} +a^8 z^2-7 a^6 z^2-14 a^4 z^2-10 a^2 z^2-4 z^2+a^9 z-4 a^7 z-8 a^5 z-4 a^3 z-a z+3 a^6+5 a^4+a^2
The A2 invariant -q^{24}-q^{20}-3 q^{18}+3 q^{16}-2 q^{14}+3 q^{12}+3 q^{10}-q^8+5 q^6-4 q^4+3 q^2-1-2 q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+10 q^{120}-10 q^{118}+4 q^{116}+11 q^{114}-32 q^{112}+56 q^{110}-75 q^{108}+70 q^{106}-41 q^{104}-19 q^{102}+105 q^{100}-184 q^{98}+239 q^{96}-232 q^{94}+140 q^{92}+12 q^{90}-210 q^{88}+385 q^{86}-475 q^{84}+426 q^{82}-233 q^{80}-67 q^{78}+367 q^{76}-556 q^{74}+551 q^{72}-349 q^{70}+12 q^{68}+307 q^{66}-480 q^{64}+418 q^{62}-138 q^{60}-217 q^{58}+498 q^{56}-544 q^{54}+320 q^{52}+92 q^{50}-523 q^{48}+802 q^{46}-791 q^{44}+499 q^{42}-7 q^{40}-488 q^{38}+825 q^{36}-877 q^{34}+640 q^{32}-209 q^{30}-256 q^{28}+578 q^{26}-650 q^{24}+469 q^{22}-106 q^{20}-264 q^{18}+479 q^{16}-454 q^{14}+191 q^{12}+181 q^{10}-496 q^8+612 q^6-476 q^4+143 q^2+247-544 q^{-2} +639 q^{-4} -511 q^{-6} +234 q^{-8} +73 q^{-10} -308 q^{-12} +400 q^{-14} -354 q^{-16} +222 q^{-18} -62 q^{-20} -65 q^{-22} +127 q^{-24} -135 q^{-26} +102 q^{-28} -54 q^{-30} +17 q^{-32} +11 q^{-34} -20 q^{-36} +18 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (4, -7)
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
16 -56 128 \frac{872}{3} \frac{160}{3} -896 -\frac{4784}{3} -\frac{608}{3} -344 \frac{2048}{3} 1568 \frac{13952}{3} \frac{2560}{3} \frac{132422}{15} -\frac{8848}{15} \frac{196688}{45} \frac{1786}{9} \frac{9782}{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=-2 is the signature of K11a151. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         51 4
1        83  -5
-1       105   5
-3      119    -2
-5     109     1
-7    711      4
-9   610       -4
-11  27        5
-13 16         -5
-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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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