T(19,2)

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T(17,2).jpg

T(17,2)

T(10,3).jpg

T(10,3)

Contents

T(19,2).jpg See other torus knots

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Knot presentations

Planar diagram presentation X13,33,14,32 X33,15,34,14 X15,35,16,34 X35,17,36,16 X17,37,18,36 X37,19,38,18 X19,1,20,38 X1,21,2,20 X21,3,22,2 X3,23,4,22 X23,5,24,4 X5,25,6,24 X25,7,26,6 X7,27,8,26 X27,9,28,8 X9,29,10,28 X29,11,30,10 X11,31,12,30 X31,13,32,12
Gauss code -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code 20 22 24 26 28 30 32 34 36 38 2 4 6 8 10 12 14 16 18
Braid presentation
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Polynomial invariants

Alexander polynomial t^9-t^8+t^7-t^6+t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5} - t^{-6} + t^{-7} - t^{-8} + t^{-9}
Conway polynomial z^{18}+17 z^{16}+120 z^{14}+455 z^{12}+1001 z^{10}+1287 z^8+924 z^6+330 z^4+45 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 19, 18 }
Jones polynomial -q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}+q^9
HOMFLY-PT polynomial (db, data sources) z^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -16 z^{14} a^{-20} +560 z^{12} a^{-18} -105 z^{12} a^{-20} +1365 z^{10} a^{-18} -364 z^{10} a^{-20} +2002 z^8 a^{-18} -715 z^8 a^{-20} +1716 z^6 a^{-18} -792 z^6 a^{-20} +792 z^4 a^{-18} -462 z^4 a^{-20} +165 z^2 a^{-18} -120 z^2 a^{-20} +10 a^{-18} -9 a^{-20}
Kauffman polynomial (db, data sources) z^{18}a^{-18}+z^{18}a^{-20}+z^{17}a^{-19}+z^{17}a^{-21}-18z^{16}a^{-18}-17z^{16}a^{-20}+z^{16}a^{-22}-16z^{15}a^{-19}-15z^{15}a^{-21}+z^{15}a^{-23}+136z^{14}a^{-18}+121z^{14}a^{-20}-14z^{14}a^{-22}+z^{14}a^{-24}+105z^{13}a^{-19}+91z^{13}a^{-21}-13z^{13}a^{-23}+z^{13}a^{-25}-560z^{12}a^{-18}-469z^{12}a^{-20}+78z^{12}a^{-22}-12z^{12}a^{-24}+z^{12}a^{-26}-364z^{11}a^{-19}-286z^{11}a^{-21}+66z^{11}a^{-23}-11z^{11}a^{-25}+z^{11}a^{-27}+1365z^{10}a^{-18}+1079z^{10}a^{-20}-220z^{10}a^{-22}+55z^{10}a^{-24}-10z^{10}a^{-26}+z^{10}a^{-28}+715z^9a^{-19}+495z^9a^{-21}-165z^9a^{-23}+45z^9a^{-25}-9z^9a^{-27}+z^9a^{-29}-2002z^8a^{-18}-1507z^8a^{-20}+330z^8a^{-22}-120z^8a^{-24}+36z^8a^{-26}-8z^8a^{-28}+z^8a^{-30}-792z^7a^{-19}-462z^7a^{-21}+210z^7a^{-23}-84z^7a^{-25}+28z^7a^{-27}-7z^7a^{-29}+z^7a^{-31}+1716z^6a^{-18}+1254z^6a^{-20}-252z^6a^{-22}+126z^6a^{-24}-56z^6a^{-26}+21z^6a^{-28}-6z^6a^{-30}+z^6a^{-32}+462z^5a^{-19}+210z^5a^{-21}-126z^5a^{-23}+70z^5a^{-25}-35z^5a^{-27}+15z^5a^{-29}-5z^5a^{-31}+z^5a^{-33}-792z^4a^{-18}-582z^4a^{-20}+84z^4a^{-22}-56z^4a^{-24}+35z^4a^{-26}-20z^4a^{-28}+10z^4a^{-30}-4z^4a^{-32}+z^4a^{-34}-120z^3a^{-19}-36z^3a^{-21}+28z^3a^{-23}-21z^3a^{-25}+15z^3a^{-27}-10z^3a^{-29}+6z^3a^{-31}-3z^3a^{-33}+z^3a^{-35}+165z^2a^{-18}+129z^2a^{-20}-8z^2a^{-22}+7z^2a^{-24}-6z^2a^{-26}+5z^2a^{-28}-4z^2a^{-30}+3z^2a^{-32}-2z^2a^{-34}+z^2a^{-36}+9za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-10a^{-18}-9a^{-20}
The A2 invariant Data:T(19,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(19,2)/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: (45, 285)
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
Data:T(19,2)/V 2,1 Data:T(19,2)/V 3,1 Data:T(19,2)/V 4,1 Data:T(19,2)/V 4,2 Data:T(19,2)/V 4,3 Data:T(19,2)/V 5,1 Data:T(19,2)/V 5,2 Data:T(19,2)/V 5,3 Data:T(19,2)/V 5,4 Data:T(19,2)/V 6,1 Data:T(19,2)/V 6,2 Data:T(19,2)/V 6,3 Data:T(19,2)/V 6,4 Data:T(19,2)/V 6,5 Data:T(19,2)/V 6,6 Data:T(19,2)/V 6,7 Data:T(19,2)/V 6,8 Data:T(19,2)/V 6,9

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=18 is the signature of T(19,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
012345678910111213141516171819χ
57                   1-1
55                    0
53                 11 0
51                    0
49               11   0
47                    0
45             11     0
43                    0
41           11       0
39                    0
37         11         0
35                    0
33       11           0
31                    0
29     11             0
27                    0
25   11               0
23                    0
21  1                 1
191                   1
171                   1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=17 i=19
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
r=5 {\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}
r=7 {\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}
r=9 {\mathbb Z}_2 {\mathbb Z}
r=10 {\mathbb Z}
r=11 {\mathbb Z}_2 {\mathbb Z}
r=12 {\mathbb Z}
r=13 {\mathbb Z}_2 {\mathbb Z}
r=14 {\mathbb Z}
r=15 {\mathbb Z}_2 {\mathbb Z}
r=16 {\mathbb Z}
r=17 {\mathbb Z}_2 {\mathbb Z}
r=18 {\mathbb Z}
r=19 {\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

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T(17,2).jpg

T(17,2)

T(10,3).jpg

T(10,3)