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

Visit K11n127 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X7,12,8,13 X9,16,10,17 X2,11,3,12 X13,21,14,20 X15,8,16,9 X22,17,1,18 X6,19,7,20 X21,15,22,14
Gauss code 1, -6, 2, -1, 3, -10, -4, 8, -5, -2, 6, 4, -7, 11, -8, 5, 9, -3, 10, 7, -11, -9
Dowker-Thistlethwaite code 4 10 18 -12 -16 2 -20 -8 22 6 -14
A Braid Representative
A Morse Link Presentation K11n127 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n127/ThurstonBennequinNumber
Hyperbolic Volume 13.129
A-Polynomial See Data:K11n127/A-polynomial

[edit Notes for K11n127's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11n127's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 55, -2 }
Jones polynomial -1+4 q^{-1} -6 q^{-2} +9 q^{-3} -9 q^{-4} +9 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-5 z^2 a^6-4 a^6+z^6 a^4+4 z^4 a^4+7 z^2 a^4+4 a^4-z^4 a^2-z^2 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+9 z^3 a^9-2 z a^9+3 z^8 a^8-7 z^6 a^8+z^4 a^8+z^2 a^8+a^8+z^9 a^7+4 z^7 a^7-18 z^5 a^7+16 z^3 a^7-6 z a^7+5 z^8 a^6-11 z^6 a^6+8 z^4 a^6-8 z^2 a^6+4 a^6+z^9 a^5+2 z^7 a^5-6 z^5 a^5+6 z^3 a^5-4 z a^5+2 z^8 a^4-3 z^6 a^4+8 z^4 a^4-9 z^2 a^4+4 a^4+z^7 a^3+2 z^5 a^3+4 z^4 a^2-2 z^2 a^2+z^3 a
The A2 invariant q^{28}-q^{24}+q^{22}-3 q^{20}-q^{18}-q^{14}+3 q^{12}+3 q^8+q^6-q^4+2 q^2-1
The G2 invariant Data:K11n127/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_31, K11n11, K11n22, K11n112,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n22,}

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{124}{3} \frac{20}{3} -128 -\frac{352}{3} -\frac{160}{3} 16 \frac{256}{3} 128 \frac{992}{3} \frac{160}{3} \frac{5071}{15} \frac{3956}{15} -\frac{2516}{45} -\frac{559}{9} -\frac{209}{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 K11n127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         1-1
-1        3 3
-3       42 -2
-5      52  3
-7     44   0
-9    55    0
-11   34     1
-13  25      -3
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\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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