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

Visit K11n74 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X16,12,17,11 X13,6,14,7 X18,16,19,15 X12,18,13,17 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 K11n74 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n74/ThurstonBennequinNumber
Hyperbolic Volume 10.4045
A-Polynomial See Data:K11n74/A-polynomial

[edit Notes for K11n74's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n74's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-2 t+3-2 t^{-1} + t^{-2}
Conway polynomial z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 9, 0 }
Jones polynomial q^6-2 q^5+2 q^4-3 q^3+2 q^2-q+2+ q^{-1} - q^{-2} + q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} +z^6-a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +7 z^4-4 a^2 z^2-12 z^2 a^{-2} +3 z^2 a^{-4} +15 z^2-4 a^2-8 a^{-2} +2 a^{-4} +11
Kauffman polynomial (db, data sources) a^2 z^8+z^8 a^{-2} +z^8 a^{-4} +z^8+a^3 z^7+2 a z^7+2 z^7 a^{-1} +3 z^7 a^{-3} +2 z^7 a^{-5} -6 a^2 z^6-6 z^6 a^{-2} -3 z^6 a^{-4} +z^6 a^{-6} -8 z^6-6 a^3 z^5-14 a z^5-15 z^5 a^{-1} -16 z^5 a^{-3} -9 z^5 a^{-5} +10 a^2 z^4+11 z^4 a^{-2} -z^4 a^{-4} -4 z^4 a^{-6} +18 z^4+10 a^3 z^3+25 a z^3+29 z^3 a^{-1} +24 z^3 a^{-3} +10 z^3 a^{-5} -8 a^2 z^2-13 z^2 a^{-2} +z^2 a^{-4} +3 z^2 a^{-6} -19 z^2-5 a^3 z-13 a z-17 z a^{-1} -13 z a^{-3} -4 z a^{-5} +4 a^2+8 a^{-2} +2 a^{-4} +11
The A2 invariant Data:K11n74/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n74/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_20, 10_140, K11n73,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n73,}

Vassiliev invariants

V2 and V3: (2, -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
8 -16 32 \frac{76}{3} \frac{20}{3} -128 -\frac{544}{3} \frac{32}{3} -48 \frac{256}{3} 128 \frac{608}{3} \frac{160}{3} \frac{6391}{15} -\frac{308}{5} \frac{12604}{45} \frac{233}{9} \frac{631}{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=0 is the signature of K11n74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          1 -1
9         11 0
7        21  -1
5      111   -1
3      12    1
1    131     1
-1   1 2      3
-3   11       0
-5 11         0
-7            0
-91           -1
Integral Khovanov Homology

(db, data source)

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