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

Visit K11n75 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X11,16,12,17 X13,6,14,7 X15,18,16,19 X17,12,18,13 X19,22,20,1 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 8, -10, 5, -11, 10
Dowker-Thistlethwaite code 4 8 -14 2 -20 -16 -6 -18 -12 -22 -10
A Braid Representative
A Morse Link Presentation K11n75 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n75/ThurstonBennequinNumber
Hyperbolic Volume 13.5931
A-Polynomial See Data:K11n75/A-polynomial

[edit Notes for K11n75's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 2

[edit Notes for K11n75's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+5 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 63, -2 }
Jones polynomial -2+5 q^{-1} -7 q^{-2} +11 q^{-3} -10 q^{-4} +10 q^{-5} -9 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+2 a^8-3 z^4 a^6-10 z^2 a^6-9 a^6+2 z^6 a^4+10 z^4 a^4+18 z^2 a^4+11 a^4-2 z^4 a^2-5 z^2 a^2-3 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+10 z^3 a^9-4 z a^9+3 z^8 a^8-6 z^6 a^8-z^4 a^8+z^2 a^8+2 a^8+z^9 a^7+7 z^7 a^7-31 z^5 a^7+36 z^3 a^7-17 z a^7+7 z^8 a^6-18 z^6 a^6+17 z^4 a^6-15 z^2 a^6+9 a^6+z^9 a^5+8 z^7 a^5-31 z^5 a^5+42 z^3 a^5-21 z a^5+4 z^8 a^4-10 z^6 a^4+19 z^4 a^4-20 z^2 a^4+11 a^4+4 z^7 a^3-10 z^5 a^3+19 z^3 a^3-11 z a^3+z^6 a^2+4 z^4 a^2-6 z^2 a^2+3 a^2+3 z^3 a-3 z a
The A2 invariant Data:K11n75/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n75/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_65, 10_77, K11n71,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n71,}

Vassiliev invariants

V2 and V3: (4, -5)
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 -40 128 \frac{536}{3} \frac{64}{3} -640 -\frac{2416}{3} -\frac{352}{3} -104 \frac{2048}{3} 800 \frac{8576}{3} \frac{1024}{3} \frac{58622}{15} \frac{3632}{15} \frac{53888}{45} \frac{370}{9} \frac{1982}{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 K11n75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         2-2
-1        3 3
-3       53 -2
-5      62  4
-7     45   1
-9    66    0
-11   34     1
-13  26      -4
-15 13       2
-17 2        -2
-191         1
Integral Khovanov Homology

(db, data source)

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

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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