T(17,2): Difference between revisions

Revision as of 22:18, 26 August 2005

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Visit T(17,2)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	X_15,33,16,32 X_33,17,34,16 X_17,1,18,34 X_1,19,2,18 X_19,3,20,2 X_3,21,4,20 X_21,5,22,4 X_5,23,6,22 X_23,7,24,6 X_7,25,8,24 X_25,9,26,8 X_9,27,10,26 X_27,11,28,10 X_11,29,12,28 X_29,13,30,12 X_13,31,14,30 X_31,15,32,14
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

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.

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

Vassiliev invariants

V₂ and V₃

{0, 204})

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

0

45

0

43

1

0

41

0

39

1

0

37

0

35

1

0

33

0

31

1

0

29

0

27

1

0

25

0

23

1

0

21

0

19

1

17

1

15

1

Computer Talk

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

Include[ColouredJonesM.mhtml]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 19, 2005, 13:11:25)...
In[2]:=	Crossings[TorusKnot[17, 2]]
Out[2]=	17
In[3]:=	PD[TorusKnot[17, 2]]
Out[3]=	PD[X[15, 33, 16, 32], X[33, 17, 34, 16], X[17, 1, 18, 34], X[1, 19, 2, 18], X[19, 3, 20, 2], X[3, 21, 4, 20], X[21, 5, 22, 4], X[5, 23, 6, 22], X[23, 7, 24, 6], X[7, 25, 8, 24], X[25, 9, 26, 8], X[9, 27, 10, 26], X[27, 11, 28, 10], X[11, 29, 12, 28], X[29, 13, 30, 12], X[13, 31, 14, 30], X[31, 15, 32, 14]]
In[4]:=	GaussCode[TorusKnot[17, 2]]
Out[4]=	GaussCode[-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]
In[5]:=	BR[TorusKnot[17, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[17, 2]][t]
Out[6]=	-8 -7 -6 -5 -4 -3 -2 1 2 3 4 1 + t - t + t - t + t - t + t - - - t + t - t + t - t 5 6 7 8 t + t - t + t
In[7]:=	Conway[TorusKnot[17, 2]][z]
Out[7]=	2 4 6 8 10 12 14 16 1 + 36 z + 210 z + 462 z + 495 z + 286 z + 91 z + 15 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[17, 2]], KnotSignature[TorusKnot[17, 2]]}
Out[9]=	{17, 16}
In[10]:=	J=Jones[TorusKnot[17, 2]][q]
Out[10]=	8 10 11 12 13 14 15 16 17 18 19 20 q + q - q + q - q + q - q + q - q + q - q + q - 21 22 23 24 25 q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[17, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[17, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[17, 2]], Vassiliev[3][TorusKnot[17, 2]]}
Out[14]=	{0, 204}
In[15]:=	Kh[TorusKnot[17, 2]][q, t]
Out[15]=	15 17 19 2 23 3 23 4 27 5 27 6 31 7 q + q + q t + q t + q t + q t + q t + q t + 31 8 35 9 35 10 39 11 39 12 43 13 43 14 q t + q t + q t + q t + q t + q t + q t + 47 15 47 16 51 17 q t + q t + q t

@@ Line 118: / Line 118: @@
 Include[ColouredJonesM.mhtml]
 <tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[TorusKnot[17, 2]][q]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 30    32      34    36    38    66    68    70
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>NotAvailable</nowiki></pre></td></tr>
-q   + q   + 2 q   + q   + q   - q   - q   - q</nowiki></pre></td></tr>
 <tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[TorusKnot[17, 2]][a, z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                                                   2
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>NotAvailable</nowiki></pre></td></tr>
-9     z     z     z     z     z     z     z     z    8 z   z
---- + --- + --- - --- + --- - --- + --- - --- + --- - --- - --- + --- -
-16    33    31    29    27    25    23    21    19    17    32
-a     a     a     a     a     a     a     a     a     a     a     a
-2      2      2      2      2       2        2    3
-z    3 z    4 z    5 z    6 z    7 z    92 z    120 z    z
-  ---- + ---- - ---- + ---- - ---- + ---- - ----- - ------ + --- -
-28     26     24     22     20      18      16      31
-  a      a      a      a      a      a       a       a       a
-3       3       3       3       3       3    4       4
-z    6 z    10 z    15 z    21 z    28 z    84 z    z     4 z
-  ---- + ---- - ----- + ----- - ----- + ----- + ----- + --- - ---- +
-27      25      23      21      19      17     30    28
-  a      a       a       a       a       a       a      a     a
-4       4       4        4        4    5       5
-z    20 z    35 z    56 z    336 z    462 z    z     5 z
-  ----- - ----- + ----- - ----- + ------ + ------ + --- - ---- +
-24      22      20      18       16      29    27
-   a       a       a       a       a        a       a     a
-5       5        5        5    6       6       6
-z    35 z    70 z    126 z    252 z    z     6 z    21 z
-  ----- - ----- + ----- - ------ - ------ + --- - ---- + ----- -
-23      21      19       17      28    26      24
-   a       a       a       a        a       a     a       a
-6        6        6    7       7       7       7
-z    126 z    582 z    792 z    z     7 z    28 z    84 z
-  ----- + ------ - ------ - ------ + --- - ---- + ----- - ----- +
-20       18       16      27    25      23      21
-   a       a        a        a       a     a       a       a
-7    8       8       8        8        8        8
-z    330 z    z     8 z    36 z    120 z    550 z    715 z
-  ------ + ------ + --- - ---- + ----- - ------ + ------ + ------ +
-17      26    24      22      20       18       16
-   a        a       a     a       a       a        a        a
-9       9        9        9    10       10       10
-  z     9 z    45 z    165 z    220 z    z     10 z     55 z
-  --- - ---- + ----- - ------ - ------ + --- - ------ + ------ -
-23      21      19       17      24     22       20
-  a     a       a       a        a       a      a        a
-10    11       11       11       11    12       12
-z     364 z     z     11 z     66 z     78 z     z     12 z
-  ------- - ------- + --- - ------ + ------ + ------ + --- - ------ +
-16      23     21       19       17      22     20
-    a         a       a      a        a        a       a      a
-12    13       13       13    14       14       14
-z     105 z     z     13 z     14 z     z     15 z     16 z
-  ------ + ------- + --- - ------ - ------ + --- - ------ - ------ +
-16      21     19       17      20     18       16
-   a         a       a      a        a       a      a        a
-15    16    16
-  z     z     z     z
-  --- + --- + --- + ---
-17    18    16
-  a     a     a     a</nowiki></pre></td></tr>
 <tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][TorusKnot[17, 2]], Vassiliev[3][TorusKnot[17, 2]]}</nowiki></pre></td></tr>
 <tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 204}</nowiki></pre></td></tr>

T(17,2): Difference between revisions

Revision as of 22:18, 26 August 2005

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Knot presentations

Polynomial invariants

Polynomial invariants

Vassiliev invariants

Computer Talk

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