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

Visit K11n182 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X3,12,4,13 X16,5,17,6 X18,8,19,7 X14,10,15,9 X11,4,12,5 X20,13,21,14 X22,16,1,15 X2,17,3,18 X8,20,9,19 X10,21,11,22
Gauss code 1, -9, -2, 6, 3, -1, 4, -10, 5, -11, -6, 2, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 -12 16 18 14 -4 20 22 2 8 10
A Braid Representative
A Morse Link Presentation K11n182 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:K11n182/ThurstonBennequinNumber
Hyperbolic Volume 16.3897
A-Polynomial See Data:K11n182/A-polynomial

[edit Notes for K11n182's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11n182's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+11 t^2-18 t+23-18 t^{-1} +11 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 93, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+13 q^2-15 q+16-15 q^{-1} +11 q^{-2} -7 q^{-3} +3 q^{-4}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-z^6 a^{-2} +5 z^6-4 a^2 z^4-3 z^4 a^{-2} +8 z^4+a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} +4 z^2+2 a^4-3 a^2+ a^{-2} +1
Kauffman polynomial (db, data sources) 3 a z^9+3 z^9 a^{-1} +6 a^2 z^8+7 z^8 a^{-2} +13 z^8+3 a^3 z^7+2 a z^7+6 z^7 a^{-1} +7 z^7 a^{-3} -14 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -27 z^6-7 a z^5-19 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} +6 a^4 z^4+23 a^2 z^4-6 z^4 a^{-4} +23 z^4-5 a^3 z^3+2 a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -9 a^4 z^2-18 a^2 z^2+2 z^2 a^{-2} +2 z^2 a^{-4} -9 z^2+3 a^3 z+2 a z-2 z a^{-1} -z a^{-3} +2 a^4+3 a^2- a^{-2} +1
The A2 invariant q^{16}+3 q^{12}-2 q^{10}+q^8-q^6-4 q^4+3 q^2-4+4 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant Data:K11n182/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: (-3, 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
-12 16 72 82 30 -192 -\frac{992}{3} -\frac{224}{3} -80 -288 128 -984 -360 -\frac{5471}{10} -\frac{1174}{15} -\frac{4462}{15} \frac{287}{6} -\frac{671}{10}

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=0 is the signature of K11n182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         1-1
9        3 3
7       51 -4
5      83  5
3     75   -2
1    98    1
-1   78     1
-3  48      -4
-5 37       4
-7 4        -4
-93         3
Integral Khovanov Homology

(db, data source)

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