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

Visit K11a150 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,5,17,6 X20,7,21,8 X12,10,13,9 X2,11,3,12 X18,13,19,14 X22,15,1,16 X8,17,9,18 X14,19,15,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -5, 7, -10, 8, -3, 9, -7, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 10 16 20 12 2 18 22 8 14 6
A Braid Representative
A Morse Link Presentation K11a150 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a150/ThurstonBennequinNumber
Hyperbolic Volume 16.2111
A-Polynomial See Data:K11a150/A-polynomial

[edit Notes for K11a150's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for K11a150's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+13 t^2-29 t+37-29 t^{-1} +13 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 125, -4 }
Jones polynomial 1-4 q^{-1} +9 q^{-2} -13 q^{-3} +18 q^{-4} -20 q^{-5} +20 q^{-6} -17 q^{-7} +12 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}+2 z^4 a^8+3 z^2 a^8+a^8-z^6 a^6-z^4 a^6-a^6-z^6 a^4-z^4 a^4+2 z^2 a^4+2 a^4+z^4 a^2+z^2 a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+5 z^7 a^{11}-7 z^5 a^{11}+3 z^3 a^{11}-z a^{11}+6 z^8 a^{10}-8 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}+a^{10}+5 z^9 a^9-4 z^7 a^9+2 z^3 a^9+2 z^{10} a^8+8 z^8 a^8-24 z^6 a^8+25 z^4 a^8-9 z^2 a^8+a^8+11 z^9 a^7-23 z^7 a^7+18 z^5 a^7-7 z^3 a^7+2 z a^7+2 z^{10} a^6+9 z^8 a^6-31 z^6 a^6+27 z^4 a^6-9 z^2 a^6+a^6+6 z^9 a^5-10 z^7 a^5+z^5 a^5+7 z^8 a^4-17 z^6 a^4+11 z^4 a^4-5 z^2 a^4+2 a^4+4 z^7 a^3-9 z^5 a^3+4 z^3 a^3+z^6 a^2-2 z^4 a^2+z^2 a^2
The A2 invariant Data:K11a150/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a150/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (5, -12)
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
20 -96 200 \frac{1702}{3} \frac{266}{3} -1920 -3680 -608 -544 \frac{4000}{3} 4608 \frac{34040}{3} \frac{5320}{3} \frac{145759}{6} 86 \frac{89966}{9} \frac{5125}{18} \frac{7999}{6}

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 K11a150. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1           11
-1          3 -3
-3         61 5
-5        84  -4
-7       105   5
-9      108    -2
-11     1010     0
-13    710      3
-15   510       -5
-17  27        5
-19 15         -4
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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