T(19,2)

T(17,2)

T(10,3)

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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

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

Further Quantum Invariants

T(19,2) Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(19,2)"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{18}+17z^{16}+120z^{14}+455z^{12}+1001z^{10}+1287z^{8}+924z^{6}+330z^{4}+45z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 19, 18 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}+q^{9}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{18}a^{-18}+18z^{16}a^{-18}-z^{16}a^{-20}+136z^{14}a^{-18}-16z^{14}a^{-20}+560z^{12}a^{-18}-105z^{12}a^{-20}+1365z^{10}a^{-18}-364z^{10}a^{-20}+2002z^{8}a^{-18}-715z^{8}a^{-20}+1716z^{6}a^{-18}-792z^{6}a^{-20}+792z^{4}a^{-18}-462z^{4}a^{-20}+165z^{2}a^{-18}-120z^{2}a^{-20}+10a^{-18}-9a^{-20}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{18}a^{-18}+z^{18}a^{-20}+z^{17}a^{-19}+z^{17}a^{-21}-18z^{16}a^{-18}-17z^{16}a^{-20}+z^{16}a^{-22}-16z^{15}a^{-19}-15z^{15}a^{-21}+z^{15}a^{-23}+136z^{14}a^{-18}+121z^{14}a^{-20}-14z^{14}a^{-22}+z^{14}a^{-24}+105z^{13}a^{-19}+91z^{13}a^{-21}-13z^{13}a^{-23}+z^{13}a^{-25}-560z^{12}a^{-18}-469z^{12}a^{-20}+78z^{12}a^{-22}-12z^{12}a^{-24}+z^{12}a^{-26}-364z^{11}a^{-19}-286z^{11}a^{-21}+66z^{11}a^{-23}-11z^{11}a^{-25}+z^{11}a^{-27}+1365z^{10}a^{-18}+1079z^{10}a^{-20}-220z^{10}a^{-22}+55z^{10}a^{-24}-10z^{10}a^{-26}+z^{10}a^{-28}+715z^{9}a^{-19}+495z^{9}a^{-21}-165z^{9}a^{-23}+45z^{9}a^{-25}-9z^{9}a^{-27}+z^{9}a^{-29}-2002z^{8}a^{-18}-1507z^{8}a^{-20}+330z^{8}a^{-22}-120z^{8}a^{-24}+36z^{8}a^{-26}-8z^{8}a^{-28}+z^{8}a^{-30}-792z^{7}a^{-19}-462z^{7}a^{-21}+210z^{7}a^{-23}-84z^{7}a^{-25}+28z^{7}a^{-27}-7z^{7}a^{-29}+z^{7}a^{-31}+1716z^{6}a^{-18}+1254z^{6}a^{-20}-252z^{6}a^{-22}+126z^{6}a^{-24}-56z^{6}a^{-26}+21z^{6}a^{-28}-6z^{6}a^{-30}+z^{6}a^{-32}+462z^{5}a^{-19}+210z^{5}a^{-21}-126z^{5}a^{-23}+70z^{5}a^{-25}-35z^{5}a^{-27}+15z^{5}a^{-29}-5z^{5}a^{-31}+z^{5}a^{-33}-792z^{4}a^{-18}-582z^{4}a^{-20}+84z^{4}a^{-22}-56z^{4}a^{-24}+35z^{4}a^{-26}-20z^{4}a^{-28}+10z^{4}a^{-30}-4z^{4}a^{-32}+z^{4}a^{-34}-120z^{3}a^{-19}-36z^{3}a^{-21}+28z^{3}a^{-23}-21z^{3}a^{-25}+15z^{3}a^{-27}-10z^{3}a^{-29}+6z^{3}a^{-31}-3z^{3}a^{-33}+z^{3}a^{-35}+165z^{2}a^{-18}+129z^{2}a^{-20}-8z^{2}a^{-22}+7z^{2}a^{-24}-6z^{2}a^{-26}+5z^{2}a^{-28}-4z^{2}a^{-30}+3z^{2}a^{-32}-2z^{2}a^{-34}+z^{2}a^{-36}+9za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-10a^{-18}-9a^{-20}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(19,2)"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}$, $-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}+q^{9}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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

V_2,1 through V_6,9 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 $t^{r}q^{j}$ 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   \ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 χ 57                                       1 -1 55                                         0 53                                   1 1   0 51                                         0 49                               1 1       0 47                                         0 45                           1 1           0 43                                         0 41                       1 1               0 39                                         0 37                   1 1                   0 35                                         0 33               1 1                       0 31                                         0 29           1 1                           0 27                                         0 25       1 1                               0 23                                         0 21     1                                   1 19 1                                       1 17 1                                       1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top.

T(17,2) T(10,3)