T(17,2)

T(8,3)

T(19,2)

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Edit T(17,2) Further Notes and Views

Knot presentations

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

Braid presentation

Polynomial invariants

Alexander polynomial	$t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}$
Conway polynomial	$z^{16}+15z^{14}+91z^{12}+286z^{10}+495z^{8}+462z^{6}+210z^{4}+36z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 16 }
Jones polynomial	$-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}-q^{11}+q^{10}+q^{8}$
HOMFLY-PT polynomial (db, data sources)	$z^{16}a^{-16}+16z^{14}a^{-16}-z^{14}a^{-18}+105z^{12}a^{-16}-14z^{12}a^{-18}+364z^{10}a^{-16}-78z^{10}a^{-18}+715z^{8}a^{-16}-220z^{8}a^{-18}+792z^{6}a^{-16}-330z^{6}a^{-18}+462z^{4}a^{-16}-252z^{4}a^{-18}+120z^{2}a^{-16}-84z^{2}a^{-18}+9a^{-16}-8a^{-18}$
Kauffman polynomial (db, data sources)	$z^{16}a^{-16}+z^{16}a^{-18}+z^{15}a^{-17}+z^{15}a^{-19}-16z^{14}a^{-16}-15z^{14}a^{-18}+z^{14}a^{-20}-14z^{13}a^{-17}-13z^{13}a^{-19}+z^{13}a^{-21}+105z^{12}a^{-16}+92z^{12}a^{-18}-12z^{12}a^{-20}+z^{12}a^{-22}+78z^{11}a^{-17}+66z^{11}a^{-19}-11z^{11}a^{-21}+z^{11}a^{-23}-364z^{10}a^{-16}-298z^{10}a^{-18}+55z^{10}a^{-20}-10z^{10}a^{-22}+z^{10}a^{-24}-220z^{9}a^{-17}-165z^{9}a^{-19}+45z^{9}a^{-21}-9z^{9}a^{-23}+z^{9}a^{-25}+715z^{8}a^{-16}+550z^{8}a^{-18}-120z^{8}a^{-20}+36z^{8}a^{-22}-8z^{8}a^{-24}+z^{8}a^{-26}+330z^{7}a^{-17}+210z^{7}a^{-19}-84z^{7}a^{-21}+28z^{7}a^{-23}-7z^{7}a^{-25}+z^{7}a^{-27}-792z^{6}a^{-16}-582z^{6}a^{-18}+126z^{6}a^{-20}-56z^{6}a^{-22}+21z^{6}a^{-24}-6z^{6}a^{-26}+z^{6}a^{-28}-252z^{5}a^{-17}-126z^{5}a^{-19}+70z^{5}a^{-21}-35z^{5}a^{-23}+15z^{5}a^{-25}-5z^{5}a^{-27}+z^{5}a^{-29}+462z^{4}a^{-16}+336z^{4}a^{-18}-56z^{4}a^{-20}+35z^{4}a^{-22}-20z^{4}a^{-24}+10z^{4}a^{-26}-4z^{4}a^{-28}+z^{4}a^{-30}+84z^{3}a^{-17}+28z^{3}a^{-19}-21z^{3}a^{-21}+15z^{3}a^{-23}-10z^{3}a^{-25}+6z^{3}a^{-27}-3z^{3}a^{-29}+z^{3}a^{-31}-120z^{2}a^{-16}-92z^{2}a^{-18}+7z^{2}a^{-20}-6z^{2}a^{-22}+5z^{2}a^{-24}-4z^{2}a^{-26}+3z^{2}a^{-28}-2z^{2}a^{-30}+z^{2}a^{-32}-8za^{-17}-za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}+9a^{-16}+8a^{-18}$
The A2 invariant	Data:T(17,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(17,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(17,2) 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(17,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{16}+15z^{14}+91z^{12}+286z^{10}+495z^{8}+462z^{6}+210z^{4}+36z^{2}+1$

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]=

{ 17, 16 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}-q^{11}+q^{10}+q^{8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{16}a^{-16}+16z^{14}a^{-16}-z^{14}a^{-18}+105z^{12}a^{-16}-14z^{12}a^{-18}+364z^{10}a^{-16}-78z^{10}a^{-18}+715z^{8}a^{-16}-220z^{8}a^{-18}+792z^{6}a^{-16}-330z^{6}a^{-18}+462z^{4}a^{-16}-252z^{4}a^{-18}+120z^{2}a^{-16}-84z^{2}a^{-18}+9a^{-16}-8a^{-18}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{16}a^{-16}+z^{16}a^{-18}+z^{15}a^{-17}+z^{15}a^{-19}-16z^{14}a^{-16}-15z^{14}a^{-18}+z^{14}a^{-20}-14z^{13}a^{-17}-13z^{13}a^{-19}+z^{13}a^{-21}+105z^{12}a^{-16}+92z^{12}a^{-18}-12z^{12}a^{-20}+z^{12}a^{-22}+78z^{11}a^{-17}+66z^{11}a^{-19}-11z^{11}a^{-21}+z^{11}a^{-23}-364z^{10}a^{-16}-298z^{10}a^{-18}+55z^{10}a^{-20}-10z^{10}a^{-22}+z^{10}a^{-24}-220z^{9}a^{-17}-165z^{9}a^{-19}+45z^{9}a^{-21}-9z^{9}a^{-23}+z^{9}a^{-25}+715z^{8}a^{-16}+550z^{8}a^{-18}-120z^{8}a^{-20}+36z^{8}a^{-22}-8z^{8}a^{-24}+z^{8}a^{-26}+330z^{7}a^{-17}+210z^{7}a^{-19}-84z^{7}a^{-21}+28z^{7}a^{-23}-7z^{7}a^{-25}+z^{7}a^{-27}-792z^{6}a^{-16}-582z^{6}a^{-18}+126z^{6}a^{-20}-56z^{6}a^{-22}+21z^{6}a^{-24}-6z^{6}a^{-26}+z^{6}a^{-28}-252z^{5}a^{-17}-126z^{5}a^{-19}+70z^{5}a^{-21}-35z^{5}a^{-23}+15z^{5}a^{-25}-5z^{5}a^{-27}+z^{5}a^{-29}+462z^{4}a^{-16}+336z^{4}a^{-18}-56z^{4}a^{-20}+35z^{4}a^{-22}-20z^{4}a^{-24}+10z^{4}a^{-26}-4z^{4}a^{-28}+z^{4}a^{-30}+84z^{3}a^{-17}+28z^{3}a^{-19}-21z^{3}a^{-21}+15z^{3}a^{-23}-10z^{3}a^{-25}+6z^{3}a^{-27}-3z^{3}a^{-29}+z^{3}a^{-31}-120z^{2}a^{-16}-92z^{2}a^{-18}+7z^{2}a^{-20}-6z^{2}a^{-22}+5z^{2}a^{-24}-4z^{2}a^{-26}+3z^{2}a^{-28}-2z^{2}a^{-30}+z^{2}a^{-32}-8za^{-17}-za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}+9a^{-16}+8a^{-18}$

"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(17,2)"];

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]=

{ $t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}$, $-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}-q^{11}+q^{10}+q^{8}$ }

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:

(36, 204)

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(17,2)/V 2,1 Data:T(17,2)/V 3,1 Data:T(17,2)/V 4,1 Data:T(17,2)/V 4,2 Data:T(17,2)/V 4,3 Data:T(17,2)/V 5,1 Data:T(17,2)/V 5,2 Data:T(17,2)/V 5,3 Data:T(17,2)/V 5,4 Data:T(17,2)/V 6,1 Data:T(17,2)/V 6,2 Data:T(17,2)/V 6,3 Data:T(17,2)/V 6,4 Data:T(17,2)/V 6,5 Data:T(17,2)/V 6,6 Data:T(17,2)/V 6,7 Data:T(17,2)/V 6,8 Data:T(17,2)/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=$16 is the signature of T(17,2). 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 χ 51                                   1 -1 49                                     0 47                               1 1   0 45                                     0 43                           1 1       0 41                                     0 39                       1 1           0 37                                     0 35                   1 1               0 33                                     0 31               1 1                   0 29                                     0 27           1 1                       0 25                                     0 23       1 1                           0 21                                     0 19     1                               1 17 1                                   1 15 1                                   1

Integral Khovanov Homology (db, data source)

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

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(8,3) T(19,2)