T(7,6): Difference between revisions

Revision as of 18:43, 28 August 2005

T(17,3)

T(35,2)

Visit [[[:Template:KnotilusURL]] T(7,6)'s page] at Knotilus!

Visit T(7,6)'s page at the original Knot Atlas!

T(7,6) Quick Notes

T(7,6) Further Notes and Views

Knot presentations

Planar diagram presentation	X_53,65,54,64 X_42,66,43,65 X_31,67,32,66 X_20,68,21,67 X_9,69,10,68 X_43,55,44,54 X_32,56,33,55 X_21,57,22,56 X_10,58,11,57 X_69,59,70,58 X_33,45,34,44 X_22,46,23,45 X_11,47,12,46 X_70,48,1,47 X_59,49,60,48 X_23,35,24,34 X_12,36,13,35 X_1,37,2,36 X_60,38,61,37 X_49,39,50,38 X_13,25,14,24 X_2,26,3,25 X_61,27,62,26 X_50,28,51,27 X_39,29,40,28 X_3,15,4,14 X_62,16,63,15 X_51,17,52,16 X_40,18,41,17 X_29,19,30,18 X_63,5,64,4 X_52,6,53,5 X_41,7,42,6 X_30,8,31,7 X_19,9,20,8
Gauss code	-18, -22, -26, 31, 32, 33, 34, 35, -5, -9, -13, -17, -21, 26, 27, 28, 29, 30, -35, -4, -8, -12, -16, 21, 22, 23, 24, 25, -30, -34, -3, -7, -11, 16, 17, 18, 19, 20, -25, -29, -33, -2, -6, 11, 12, 13, 14, 15, -20, -24, -28, -32, -1, 6, 7, 8, 9, 10, -15, -19, -23, -27, -31, 1, 2, 3, 4, 5, -10, -14
Dowker-Thistlethwaite code	36 14 -52 -30 68 46 24 -62 -40 8 56 34 -2 -50 18 66 44 -12 -60 28 6 54 -22 -70 38 16 64 -32 -10 48 26 4 -42 -20 58
Conway Notation	Data:T(7,6)/Conway Notation

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^{15}-t^{14}+t^9-t^7+t^3-1+ t^{-3} - t^{-7} + t^{-9} - t^{-14} + t^{-15} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{30}+29 z^{28}+377 z^{26}+2900 z^{24}+14674 z^{22}+51359 z^{20}+127282 z^{18}+224826 z^{16}+281144 z^{14}+244074 z^{12}+142208 z^{10}+52844 z^8+11649 z^6+1365 z^4+70 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 7, 18 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{24}-q^{22}+q^{21}+q^{19}+q^{17}+q^{15}}
HOMFLY-PT polynomial (db, data sources)	Data:T(7,6)/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:T(7,6)/Kauffman Polynomial
The A2 invariant	Data:T(7,6)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(7,6)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(7,6) 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(7,6)"];

In[4]:= Alexander[K][t]

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^{15}-t^{14}+t^9-t^7+t^3-1+ t^{-3} - t^{-7} + t^{-9} - t^{-14} + t^{-15} }

In[5]:= Conway[K][z]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{30}+29 z^{28}+377 z^{26}+2900 z^{24}+14674 z^{22}+51359 z^{20}+127282 z^{18}+224826 z^{16}+281144 z^{14}+244074 z^{12}+142208 z^{10}+52844 z^8+11649 z^6+1365 z^4+70 z^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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 7, 18 }

In[8]:= Jones[K][q]

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

Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{24}-q^{22}+q^{21}+q^{19}+q^{17}+q^{15}}

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= Data:T(7,6)/HOMFLYPT Polynomial

In[10]:= Kauffman[K][a, z]

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

Out[10]= Data:T(7,6)/Kauffman Polynomial

Vassiliev invariants

V₂ and V₃:

(70, 490)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 18 is the signature of T(7,6). 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

0

55

1

0

53

1

2

1

-1

51

1

2

1

-1

49

3

1

-1

47

3

1

-1

45

2

1

2

-1

43

1

2

0

41

1

2

1

39

1

37

1

35

1

33

1

31

1

29

1

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[TorusKnot[7, 6]]
Out[2]=	35
In[3]:=	PD[TorusKnot[7, 6]]
Out[3]=	PD[X[53, 65, 54, 64], X[42, 66, 43, 65], X[31, 67, 32, 66], X[20, 68, 21, 67], X[9, 69, 10, 68], X[43, 55, 44, 54], X[32, 56, 33, 55], X[21, 57, 22, 56], X[10, 58, 11, 57], X[69, 59, 70, 58], X[33, 45, 34, 44], X[22, 46, 23, 45], X[11, 47, 12, 46], X[70, 48, 1, 47], X[59, 49, 60, 48], X[23, 35, 24, 34], X[12, 36, 13, 35], X[1, 37, 2, 36], X[60, 38, 61, 37], X[49, 39, 50, 38], X[13, 25, 14, 24], X[2, 26, 3, 25], X[61, 27, 62, 26], X[50, 28, 51, 27], X[39, 29, 40, 28], X[3, 15, 4, 14], X[62, 16, 63, 15], X[51, 17, 52, 16], X[40, 18, 41, 17], X[29, 19, 30, 18], X[63, 5, 64, 4], X[52, 6, 53, 5], X[41, 7, 42, 6], X[30, 8, 31, 7], X[19, 9, 20, 8]]
In[4]:=	GaussCode[TorusKnot[7, 6]]
Out[4]=	GaussCode[-18, -22, -26, 31, 32, 33, 34, 35, -5, -9, -13, -17, -21, 26, 27, 28, 29, 30, -35, -4, -8, -12, -16, 21, 22, 23, 24, 25, -30, -34, -3, -7, -11, 16, 17, 18, 19, 20, -25, -29, -33, -2, -6, 11, 12, 13, 14, 15, -20, -24, -28, -32, -1, 6, 7, 8, 9, 10, -15, -19, -23, -27, -31, 1, 2, 3, 4, 5, -10, -14]
In[5]:=	BR[TorusKnot[7, 6]]
Out[5]=	BR[6, {1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5}]
In[6]:=	alex = Alexander[TorusKnot[7, 6]][t]
Out[6]=	-15 -14 -9 -7 -3 3 7 9 14 15 -1 + t - t + t - t + t + t - t + t - t + t
In[7]:=	Conway[TorusKnot[7, 6]][z]
Out[7]=	2 4 6 8 10 12 1 + 70 z + 1365 z + 11649 z + 52844 z + 142208 z + 244074 z + 14 16 18 20 22 281144 z + 224826 z + 127282 z + 51359 z + 14674 z + 24 26 28 30 2900 z + 377 z + 29 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[7, 6]], KnotSignature[TorusKnot[7, 6]]}
Out[9]=	{7, 18}
In[10]:=	J=Jones[TorusKnot[7, 6]][q]
Out[10]=	15 17 19 21 22 24 26 q + q + q + q - q - q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[7, 6]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[7, 6]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[7, 6]], Vassiliev[3][TorusKnot[7, 6]]}
Out[14]=	{0, 490}
In[15]:=	Kh[TorusKnot[7, 6]][q, t]
Out[15]=	29 31 33 2 37 3 35 4 37 4 39 5 41 5 q + q + q t + q t + q t + q t + q t + q t + 37 6 39 6 41 7 43 7 39 8 41 8 43 9 q t + q t + q t + q t + q t + 2 q t + q t + 45 9 41 10 43 10 45 11 47 11 45 12 2 q t + q t + 2 q t + q t + 3 q t + 2 q t + 47 12 51 12 49 13 51 13 47 14 49 14 q t + q t + 3 q t + q t + q t + q t + 53 14 51 15 53 15 49 16 51 16 55 16 q t + 2 q t + 2 q t + q t + q t + q t + 57 16 53 17 55 17 53 18 57 19 q t + q t + q t + q t + q t

This category should contain all the individual knots pages, like 7_5, K11n67, L8a2 and T(5,3)

@@ Line 1: / Line 1: @@
-<!-- Script generated - do not edit! -->
+<!--  -->
 <!--  -->
@@ Line 10: / Line 10: @@
 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-18,-22,-26,31,32,33,34,35,-5,-9,-13,-17,-21,26,27,28,29,30,-35,-4,-8,-12,-16,21,22,23,24,25,-30,-34,-3,-7,-11,16,17,18,19,20,-25,-29,-33,-2,-6,11,12,13,14,15,-20,-24,-28,-32,-1,6,7,8,9,10,-15,-19,-23,-27,-31,1,2,3,4,5,-10,-14/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=7|n=6|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-18,-22,-26,31,32,33,34,35,-5,-9,-13,-17,-21,26,27,28,29,30,-35,-4,-8,-12,-16,21,22,23,24,25,-30,-34,-3,-7,-11,16,17,18,19,20,-25,-29,-33,-2,-6,11,12,13,14,15,-20,-24,-28,-32,-1,6,7,8,9,10,-15,-19,-23,-27,-31,1,2,3,4,5,-10,-14/goTop.html}}
-Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/7.6.html {{PAGENAME}}'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!
 {{:{{PAGENAME}} Quick Notes}}
@@ Line 22: / Line 20: @@
 {{Knot Presentations}}
-===Knot presentations===
-{|
-|'''[[Planar Diagrams|Planar diagram presentation]]'''
-|style="padding-left: 1em;" | X<sub>53,65,54,64</sub> X<sub>42,66,43,65</sub> X<sub>31,67,32,66</sub> X<sub>20,68,21,67</sub> X<sub>9,69,10,68</sub> X<sub>43,55,44,54</sub> X<sub>32,56,33,55</sub> X<sub>21,57,22,56</sub> X<sub>10,58,11,57</sub> X<sub>69,59,70,58</sub> X<sub>33,45,34,44</sub> X<sub>22,46,23,45</sub> X<sub>11,47,12,46</sub> X<sub>70,48,1,47</sub> X<sub>59,49,60,48</sub> X<sub>23,35,24,34</sub> X<sub>12,36,13,35</sub> X<sub>1,37,2,36</sub> X<sub>60,38,61,37</sub> X<sub>49,39,50,38</sub> X<sub>13,25,14,24</sub> X<sub>2,26,3,25</sub> X<sub>61,27,62,26</sub> X<sub>50,28,51,27</sub> X<sub>39,29,40,28</sub> X<sub>3,15,4,14</sub> X<sub>62,16,63,15</sub> X<sub>51,17,52,16</sub> X<sub>40,18,41,17</sub> X<sub>29,19,30,18</sub> X<sub>63,5,64,4</sub> X<sub>52,6,53,5</sub> X<sub>41,7,42,6</sub> X<sub>30,8,31,7</sub> X<sub>19,9,20,8</sub>
-|-
-|'''[[Gauss Codes|Gauss code]]'''
-|style="padding-left: 1em;" | <math>\{-18,-22,-26,31,32,33,34,35,-5,-9,-13,-17,-21,26,27,28,29,30,-35,-4,-8,-12,-16,21,22,23,24,25,-30,-34,-3,-7,-11,16,17,18,19,20,-25,-29,-33,-2,-6,11,12,13,14,15,-20,-24,-28,-32,-1,6,7,8,9,10,-15,-19,-23,-27,-31,1,2,3,4,5,-10,-14\}</math>
-|-
-|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
-|style="padding-left: 1em;" | 36 14 -52 -30 68 46 24 -62 -40 8 56 34 -2 -50 18 66 44 -12 -60 28 6 54 -22 -70 38 16 64 -32 -10 48 26 4 -42 -20 58
-|}
 {{Polynomial Invariants}}
 {{Vassiliev Invariants}}
@@ Line 163: / Line 147: @@
   q   t   + q   t   + q   t   + q   t   + q   t</nowiki></pre></td></tr>
 </table>
+ {{Category:Knot Page}}

T(7,6): Difference between revisions

Revision as of 18:43, 28 August 2005

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Knot presentations

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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