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

Visit K11a162 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X14,7,15,8 X16,10,17,9 X2,11,3,12 X6,13,7,14 X20,16,21,15 X22,17,1,18 X12,20,13,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -7, 4, -11, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 18 14 16 2 6 20 22 12 8
A Braid Representative
A Morse Link Presentation K11a162 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a162/ThurstonBennequinNumber
Hyperbolic Volume 18.245
A-Polynomial See Data:K11a162/A-polynomial

[edit Notes for K11a162's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a162's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+7 t^3-20 t^2+35 t-41+35 t^{-1} -20 t^{-2} +7 t^{-3} - t^{-4}
Conway polynomial -z^8-z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 167, -2 }
Jones polynomial q^3-5 q^2+11 q-17+24 q^{-1} -27 q^{-2} +27 q^{-3} -23 q^{-4} +17 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-5 a^2 z^4+2 z^4-2 a^6 z^2+5 a^4 z^2-a^2 z^2-a^6+a^4+a^2
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+18 a^3 z^9+9 a z^9+12 a^6 z^8+20 a^4 z^8+18 a^2 z^8+10 z^8+9 a^7 z^7-2 a^5 z^7-27 a^3 z^7-11 a z^7+5 z^7 a^{-1} +4 a^8 z^6-18 a^6 z^6-53 a^4 z^6-54 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-13 a^7 z^5-18 a^5 z^5-2 a^3 z^5-7 a z^5-9 z^5 a^{-1} -4 a^8 z^4+13 a^6 z^4+42 a^4 z^4+39 a^2 z^4-z^4 a^{-2} +13 z^4-a^9 z^3+9 a^7 z^3+18 a^5 z^3+13 a^3 z^3+8 a z^3+3 z^3 a^{-1} +a^8 z^2-6 a^6 z^2-13 a^4 z^2-7 a^2 z^2-z^2-3 a^7 z-5 a^5 z-3 a^3 z-a z+a^6+a^4-a^2
The A2 invariant -q^{24}+q^{22}-3 q^{18}+4 q^{16}-4 q^{14}+2 q^{12}+2 q^{10}-3 q^8+6 q^6-5 q^4+5 q^2-2 q^{-2} +3 q^{-4} -3 q^{-6} + q^{-8}
The G2 invariant q^{128}-3 q^{126}+7 q^{124}-13 q^{122}+17 q^{120}-19 q^{118}+12 q^{116}+11 q^{114}-46 q^{112}+94 q^{110}-138 q^{108}+151 q^{106}-118 q^{104}+17 q^{102}+152 q^{100}-344 q^{98}+509 q^{96}-563 q^{94}+427 q^{92}-103 q^{90}-372 q^{88}+858 q^{86}-1159 q^{84}+1122 q^{82}-679 q^{80}-70 q^{78}+863 q^{76}-1399 q^{74}+1437 q^{72}-936 q^{70}+69 q^{68}+791 q^{66}-1265 q^{64}+1134 q^{62}-429 q^{60}-505 q^{58}+1243 q^{56}-1419 q^{54}+900 q^{52}+100 q^{50}-1198 q^{48}+1938 q^{46}-1983 q^{44}+1306 q^{42}-122 q^{40}-1126 q^{38}+1989 q^{36}-2169 q^{34}+1607 q^{32}-550 q^{30}-614 q^{28}+1451 q^{26}-1645 q^{24}+1181 q^{22}-254 q^{20}-695 q^{18}+1248 q^{16}-1177 q^{14}+500 q^{12}+447 q^{10}-1235 q^8+1534 q^6-1190 q^4+367 q^2+595-1310 q^{-2} +1523 q^{-4} -1208 q^{-6} +535 q^{-8} +190 q^{-10} -721 q^{-12} +919 q^{-14} -794 q^{-16} +480 q^{-18} -114 q^{-20} -162 q^{-22} +286 q^{-24} -290 q^{-26} +206 q^{-28} -102 q^{-30} +23 q^{-32} +28 q^{-34} -42 q^{-36} +36 q^{-38} -24 q^{-40} +11 q^{-42} -4 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (2, -3)
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 -24 32 \frac{220}{3} -\frac{4}{3} -192 -304 0 -56 \frac{256}{3} 288 \frac{1760}{3} -\frac{32}{3} \frac{18991}{15} -\frac{724}{15} \frac{16324}{45} \frac{833}{9} \frac{271}{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 K11a162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          4 -4
3         71 6
1        104  -6
-1       147   7
-3      1411    -3
-5     1313     0
-7    1014      4
-9   713       -6
-11  310        7
-13 17         -6
-15 3          3
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\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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