T(19,2): Difference between revisions

Revision as of 22:18, 26 August 2005

[[Image:T(17,2).{{{ext}}}|80px|link=T(17,2)]]

[[Image:T(10,3).{{{ext}}}|80px|link=T(10,3)]]

Visit T(19,2)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	X_13,33,14,32 X_33,15,34,14 X_15,35,16,34 X_35,17,36,16 X_17,37,18,36 X_37,19,38,18 X_19,1,20,38 X_1,21,2,20 X_21,3,22,2 X_3,23,4,22 X_23,5,24,4 X_5,25,6,24 X_25,7,26,6 X_7,27,8,26 X_27,9,28,8 X_9,29,10,28 X_29,11,30,10 X_11,31,12,30 X_31,13,32,12
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

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.

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

Vassiliev invariants

V₂ and V₃

{0, 285})

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

0

51

0

49

1

0

47

0

45

1

0

43

0

41

1

0

39

0

37

1

0

35

0

33

1

0

31

0

29

1

0

27

0

25

1

0

23

0

21

1

19

1

17

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 19, 2005, 13:11:25)...
In[2]:=	Crossings[TorusKnot[19, 2]]
Out[2]=	19
In[3]:=	PD[TorusKnot[19, 2]]
Out[3]=	PD[X[13, 33, 14, 32], X[33, 15, 34, 14], X[15, 35, 16, 34], X[35, 17, 36, 16], X[17, 37, 18, 36], X[37, 19, 38, 18], X[19, 1, 20, 38], X[1, 21, 2, 20], X[21, 3, 22, 2], X[3, 23, 4, 22], X[23, 5, 24, 4], X[5, 25, 6, 24], X[25, 7, 26, 6], X[7, 27, 8, 26], X[27, 9, 28, 8], X[9, 29, 10, 28], X[29, 11, 30, 10], X[11, 31, 12, 30], X[31, 13, 32, 12]]
In[4]:=	GaussCode[TorusKnot[19, 2]]
Out[4]=	GaussCode[-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]
In[5]:=	BR[TorusKnot[19, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[19, 2]][t]
Out[6]=	-9 -8 -7 -6 -5 -4 -3 -2 1 2 3 -1 + t - t + t - t + t - t + t - t + - + t - t + t - t 4 5 6 7 8 9 t + t - t + t - t + t
In[7]:=	Conway[TorusKnot[19, 2]][z]
Out[7]=	2 4 6 8 10 12 14 1 + 45 z + 330 z + 924 z + 1287 z + 1001 z + 455 z + 120 z + 16 18 17 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[19, 2]], KnotSignature[TorusKnot[19, 2]]}
Out[9]=	{19, 18}
In[10]:=	J=Jones[TorusKnot[19, 2]][q]
Out[10]=	9 11 12 13 14 15 16 17 18 19 20 21 q + q - q + q - q + q - q + q - q + q - q + q - 22 23 24 25 26 27 28 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[19, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[19, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[19, 2]], Vassiliev[3][TorusKnot[19, 2]]}
Out[14]=	{0, 285}
In[15]:=	Kh[TorusKnot[19, 2]][q, t]
Out[15]=	17 19 21 2 25 3 25 4 29 5 29 6 33 7 q + q + q t + q t + q t + q t + q t + q t + 33 8 37 9 37 10 41 11 41 12 45 13 45 14 q t + q t + q t + q t + q t + q t + q t + 49 15 49 16 53 17 53 18 57 19 q t + q t + q t + q t + q t

@@ Line 125: / Line 125: @@
 <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[19, 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> 34    36      38    40    42    74    76    78
+<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[19, 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>-9    10     z     z     z     z     z     z     z     z     z    9 z
+<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>
---- - --- + --- - --- + --- - --- + --- - --- + --- - --- + --- + --- +
-18    37    35    33    31    29    27    25    23    21    19
-a     a     a     a     a     a     a     a     a     a     a     a
-2      2      2      2      2      2      2        2
-  z     2 z    3 z    4 z    5 z    6 z    7 z    8 z    129 z
-  --- - ---- + ---- - ---- + ---- - ---- + ---- - ---- + ------ +
-34     32     30     28     26     24     22      20
-  a     a      a      a      a      a      a      a       a
-3       3      3       3       3       3       3       3
-z    z     3 z    6 z    10 z    15 z    21 z    28 z    36 z
-  ------ + --- - ---- + ---- - ----- + ----- - ----- + ----- - ----- -
-35    33     31      29      27      25      23      21
-   a       a     a      a       a       a       a       a       a
-4       4       4       4       4       4       4
-z    z     4 z    10 z    20 z    35 z    56 z    84 z
-  ------ + --- - ---- + ----- - ----- + ----- - ----- + ----- -
-34    32      30      28      26      24      22
-   a       a     a       a       a       a       a       a
-4    5       5       5       5       5        5
-z    792 z    z     5 z    15 z    35 z    70 z    126 z
-  ------ - ------ + --- - ---- + ----- - ----- + ----- - ------ +
-18      33    31      29      27      25      23
-   a        a       a     a       a       a       a       a
-5    6       6       6       6        6        6
-z    462 z    z     6 z    21 z    56 z    126 z    252 z
-  ------ + ------ + --- - ---- + ----- - ----- + ------ - ------ +
-19      32    30      28      26      24       22
-   a        a       a     a       a       a       a        a
-6    7       7       7       7        7        7
-z    1716 z    z     7 z    28 z    84 z    210 z    462 z
-  ------- + ------- + --- - ---- + ----- - ----- + ------ - ------ -
-18      31    29      27      25      23       21
-    a         a       a     a       a       a       a        a
-8       8       8        8        8         8         8
-z    z     8 z    36 z    120 z    330 z    1507 z    2002 z
-  ------ + --- - ---- + ----- - ------ + ------ - ------- - ------- +
-30    28      26      24       22        20        18
-   a       a     a       a       a        a         a         a
-9       9        9        9        9    10       10
-  z     9 z    45 z    165 z    495 z    715 z    z     10 z
-  --- - ---- + ----- - ------ + ------ + ------ + --- - ------ +
-27      25      23       21       19      28     26
-  a     a       a       a        a        a       a      a
-10         10         10    11       11       11
-z     220 z     1079 z     1365 z     z     11 z     66 z
-  ------ - ------- + -------- + -------- + --- - ------ + ------ -
-22        20         18       27     25       23
-   a         a         a          a        a      a        a
-11    12       12       12        12        12    13
-z     364 z     z     12 z     78 z     469 z     560 z     z
-  ------- - ------- + --- - ------ + ------ - ------- - ------- + --- -
-19      26     24       22        20        18      25
-    a         a       a      a        a         a         a       a
-13        13    14       14        14        14    15
-z     91 z     105 z     z     14 z     121 z     136 z     z
-  ------ + ------ + ------- + --- - ------ + ------- + ------- + --- -
-21        19      24     22        20        18      23
-   a        a         a       a      a         a         a       a
-15    16       16       16    17    17    18    18
-z     16 z     z     17 z     18 z     z     z     z     z
-  ------ - ------ + --- - ------ - ------ + --- + --- + --- + ---
-19      22     20       18      21    19    20    18
-   a        a       a      a        a       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[19, 2]], Vassiliev[3][TorusKnot[19, 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, 285}</nowiki></pre></td></tr>

T(19,2): 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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