7 5

7_4

7_6

Visit 7 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 5's page at Knotilus! Visit 7 5's page at the original Knot Atlas!

7 5 Quick Notes

7 5 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3
Gauss code	-1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
Dowker-Thistlethwaite code	4 10 12 14 2 8 6
Conway Notation	[322]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number 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 \text{$\$$Failed}} Hyperbolic Volume 6.44354 A-Polynomial See Data:7 5/A-polynomial

[edit Notes for 7 5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,5))$ Rasmussen s-Invariant -4

[edit Notes for 7 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-4t+5-4t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, -4 }
Jones polynomial	$-q^{-9}+2q^{-8}-3q^{-7}+3q^{-6}-3q^{-5}+3q^{-4}-q^{-3}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}\left(-z^{2}\right)-a^{8}+a^{6}z^{4}+2a^{6}z^{2}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}$
Kauffman polynomial (db, data sources)	$a^{11}z^{3}-a^{11}z+2a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{8}z^{6}+a^{8}z^{2}-a^{8}+3a^{7}z^{5}-4a^{7}z^{3}+a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}-a^{5}z+a^{4}z^{4}-3a^{4}z^{2}+2a^{4}$
The A2 invariant	$-q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2q^{10}+q^{6}$
The G2 invariant	$q^{148}-q^{146}+2q^{144}-2q^{142}+q^{138}-2q^{136}+5q^{134}-5q^{132}+4q^{130}-2q^{128}-3q^{126}+4q^{124}-6q^{122}+5q^{120}-3q^{118}-q^{116}+3q^{114}-3q^{112}+q^{110}+2q^{108}-5q^{106}+4q^{104}-3q^{102}-2q^{100}+5q^{98}-7q^{96}+8q^{94}-7q^{92}+2q^{90}+2q^{88}-6q^{86}+6q^{84}-7q^{82}+4q^{80}-2q^{76}+3q^{74}-3q^{72}+2q^{70}+3q^{68}-5q^{66}+3q^{64}-2q^{60}+7q^{58}-5q^{56}+5q^{54}-q^{52}+4q^{48}-4q^{46}+5q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 7_5.

A1 Invariants.

Weight	Invariant
1	$-q^{19}+q^{17}-q^{15}+2q^{7}+q^{3}$
2	$q^{52}-q^{50}-q^{48}+3q^{46}-q^{44}-3q^{42}+3q^{40}-2q^{36}+q^{34}-q^{30}-2q^{28}+q^{26}+q^{24}-3q^{22}+q^{20}+3q^{18}-2q^{16}+q^{14}+3q^{12}+q^{6}$
3	$-q^{99}+q^{97}+q^{95}-q^{93}-2q^{91}+q^{89}+5q^{87}-q^{85}-7q^{83}-q^{81}+7q^{79}+3q^{77}-7q^{75}-3q^{73}+7q^{71}+4q^{69}-4q^{67}-3q^{65}+2q^{63}+2q^{61}-3q^{57}-2q^{55}+q^{53}+4q^{51}-2q^{49}-6q^{47}+7q^{43}-q^{41}-7q^{39}-2q^{37}+6q^{35}+3q^{33}-6q^{31}-3q^{29}+4q^{27}+5q^{25}-q^{23}-2q^{21}+q^{19}+3q^{17}+q^{15}+q^{9}$
4	$q^{160}-q^{158}-q^{156}+q^{154}+2q^{150}-3q^{148}-3q^{146}+2q^{144}+3q^{142}+9q^{140}-5q^{138}-12q^{136}-4q^{134}+5q^{132}+19q^{130}-15q^{126}-13q^{124}+24q^{120}+8q^{118}-12q^{116}-16q^{114}-5q^{112}+17q^{110}+9q^{108}-4q^{106}-10q^{104}-6q^{102}+7q^{100}+7q^{98}+3q^{96}-2q^{94}-5q^{92}-3q^{90}+4q^{88}+9q^{86}+q^{84}-7q^{82}-12q^{80}+2q^{78}+13q^{76}+6q^{74}-7q^{72}-20q^{70}+q^{68}+16q^{66}+10q^{64}-4q^{62}-22q^{60}-5q^{58}+11q^{56}+13q^{54}+5q^{52}-17q^{50}-10q^{48}+2q^{46}+10q^{44}+10q^{42}-6q^{40}-7q^{38}-4q^{36}+3q^{34}+8q^{32}+q^{30}-2q^{26}+3q^{22}+q^{20}+q^{18}+q^{12}$
5	$-q^{235}+q^{233}+q^{231}-q^{229}+q^{221}+2q^{219}-2q^{217}-5q^{215}-3q^{213}+q^{211}+8q^{209}+10q^{207}+4q^{205}-12q^{203}-21q^{201}-9q^{199}+12q^{197}+29q^{195}+22q^{193}-7q^{191}-37q^{189}-36q^{187}+38q^{183}+46q^{181}+12q^{179}-34q^{177}-53q^{175}-23q^{173}+29q^{171}+51q^{169}+29q^{167}-18q^{165}-45q^{163}-30q^{161}+8q^{159}+35q^{157}+27q^{155}-2q^{153}-23q^{151}-24q^{149}-5q^{147}+13q^{145}+17q^{143}+8q^{141}-3q^{139}-12q^{137}-13q^{135}-2q^{133}+10q^{131}+16q^{129}+12q^{127}-7q^{125}-21q^{123}-14q^{121}+7q^{119}+27q^{117}+23q^{115}-7q^{113}-34q^{111}-26q^{109}+4q^{107}+36q^{105}+37q^{103}-3q^{101}-41q^{99}-41q^{97}-4q^{95}+37q^{93}+45q^{91}+13q^{89}-34q^{87}-47q^{85}-22q^{83}+20q^{81}+41q^{79}+29q^{77}-6q^{75}-36q^{73}-32q^{71}-4q^{69}+21q^{67}+29q^{65}+14q^{63}-11q^{61}-23q^{59}-16q^{57}-q^{55}+13q^{53}+15q^{51}+5q^{49}-4q^{47}-9q^{45}-6q^{43}+q^{41}+6q^{39}+4q^{37}+3q^{35}-2q^{31}+2q^{27}+q^{25}+q^{23}+q^{21}+q^{15}$
6	$q^{324}-q^{322}-q^{320}+q^{318}-2q^{312}+2q^{310}-2q^{306}+4q^{304}+3q^{302}+q^{300}-5q^{298}-2q^{296}-8q^{294}-8q^{292}+8q^{290}+16q^{288}+17q^{286}+q^{284}-4q^{282}-31q^{280}-39q^{278}-6q^{276}+28q^{274}+55q^{272}+41q^{270}+19q^{268}-49q^{266}-90q^{264}-61q^{262}+4q^{260}+78q^{258}+100q^{256}+84q^{254}-26q^{252}-117q^{250}-125q^{248}-58q^{246}+54q^{244}+123q^{242}+142q^{240}+32q^{238}-89q^{236}-140q^{234}-104q^{232}+2q^{230}+92q^{228}+142q^{226}+68q^{224}-34q^{222}-100q^{220}-98q^{218}-33q^{216}+37q^{214}+93q^{212}+63q^{210}+6q^{208}-44q^{206}-60q^{204}-38q^{202}-2q^{200}+39q^{198}+40q^{196}+24q^{194}-4q^{192}-24q^{190}-32q^{188}-23q^{186}+q^{184}+22q^{182}+35q^{180}+21q^{178}-5q^{176}-37q^{174}-39q^{172}-21q^{170}+23q^{168}+58q^{166}+45q^{164}+4q^{162}-54q^{160}-66q^{158}-43q^{156}+33q^{154}+92q^{152}+74q^{150}+11q^{148}-73q^{146}-100q^{144}-76q^{142}+30q^{140}+115q^{138}+109q^{136}+38q^{134}-67q^{132}-119q^{130}-115q^{128}-4q^{126}+102q^{124}+127q^{122}+79q^{120}-22q^{118}-98q^{116}-134q^{114}-57q^{112}+42q^{110}+101q^{108}+100q^{106}+39q^{104}-33q^{102}-105q^{100}-82q^{98}-24q^{96}+37q^{94}+73q^{92}+65q^{90}+28q^{88}-40q^{86}-56q^{84}-47q^{82}-17q^{80}+18q^{78}+41q^{76}+40q^{74}+6q^{72}-10q^{70}-25q^{68}-24q^{66}-13q^{64}+7q^{62}+19q^{60}+10q^{58}+8q^{56}-q^{54}-7q^{52}-9q^{50}-2q^{48}+4q^{46}+2q^{44}+5q^{42}+3q^{40}+q^{38}-2q^{36}+2q^{32}+q^{28}+q^{26}+q^{24}+q^{18}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2q^{10}+q^{6}$
1,1	$q^{76}-2q^{74}+4q^{72}-6q^{70}+9q^{68}-12q^{66}+14q^{64}-14q^{62}+13q^{60}-10q^{58}+2q^{56}+4q^{54}-12q^{52}+20q^{50}-22q^{48}+28q^{46}-26q^{44}+22q^{42}-22q^{40}+8q^{38}-10q^{36}-4q^{34}+6q^{32}-12q^{30}+16q^{28}-10q^{26}+16q^{24}-6q^{22}+10q^{20}-2q^{18}+4q^{16}+q^{12}$
2,0	$q^{70}+q^{62}-2q^{58}+q^{54}+q^{48}-3q^{44}-3q^{42}-2q^{40}-3q^{38}-2q^{36}+q^{34}+q^{32}+3q^{28}+3q^{26}+2q^{24}+q^{22}+3q^{20}+2q^{18}+q^{12}$
3,0	$-q^{126}-q^{116}+2q^{114}+2q^{112}+2q^{110}-2q^{108}-5q^{106}-q^{104}+2q^{102}+3q^{100}-2q^{98}-3q^{96}+2q^{94}+6q^{92}+4q^{90}-2q^{88}-2q^{86}+2q^{84}+5q^{82}+q^{80}-3q^{78}-q^{76}+q^{74}-4q^{70}-4q^{68}-q^{66}-2q^{64}-6q^{62}-6q^{60}-3q^{58}+q^{56}-3q^{54}-5q^{52}-2q^{50}+5q^{48}+6q^{46}-q^{42}+2q^{40}+8q^{38}+6q^{36}+q^{34}-q^{32}+2q^{30}+4q^{28}+3q^{26}+q^{18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{62}-q^{60}+2q^{56}-2q^{54}+3q^{50}-2q^{48}-q^{46}-2q^{42}-2q^{40}-q^{38}-q^{34}-2q^{32}+q^{30}+q^{28}-2q^{26}+4q^{24}+3q^{22}+q^{20}+3q^{18}+2q^{16}+q^{12}$
1,0,0	$-q^{37}-q^{33}-q^{29}-q^{25}+q^{21}+q^{19}+2q^{17}+q^{15}+2q^{13}+q^{9}$
1,0,1	$q^{100}-2q^{98}+3q^{96}-q^{94}-3q^{92}+8q^{90}-10q^{88}+6q^{86}+3q^{84}-12q^{82}+18q^{80}-15q^{78}+4q^{76}+6q^{74}-17q^{72}+17q^{70}-10q^{68}+3q^{66}+9q^{64}-2q^{62}+3q^{60}+6q^{58}-6q^{56}-7q^{54}+5q^{52}-26q^{50}+6q^{48}-14q^{46}-12q^{44}+13q^{42}-17q^{40}+17q^{38}+3q^{36}+3q^{34}+15q^{32}-q^{30}+10q^{28}+4q^{26}+2q^{24}+4q^{22}+q^{18}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{80}-q^{76}+q^{74}+2q^{72}+2q^{66}+q^{64}-2q^{62}-2q^{60}-q^{58}-4q^{56}-5q^{54}-q^{52}-2q^{50}-3q^{48}+q^{44}-2q^{42}-q^{40}+2q^{38}+2q^{36}+q^{34}+3q^{32}+6q^{30}+3q^{28}+3q^{26}+3q^{24}+2q^{22}+q^{18}$
1,0,0,0	$-q^{46}-q^{42}-q^{40}-q^{36}-q^{32}+q^{26}+q^{24}+2q^{22}+2q^{20}+q^{18}+2q^{16}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{62}+q^{60}-2q^{58}+2q^{56}-2q^{54}+2q^{52}-q^{50}+q^{46}-2q^{44}+2q^{42}-4q^{40}+3q^{38}-4q^{36}+3q^{34}-2q^{32}+q^{30}+q^{28}+2q^{24}-q^{22}+3q^{20}-q^{18}+2q^{16}+q^{12}$
1,0	$q^{100}-q^{96}-q^{94}+q^{92}+2q^{90}-2q^{86}-q^{84}+2q^{82}+2q^{80}-q^{78}-2q^{76}+q^{72}-q^{70}-2q^{68}-q^{66}+q^{64}-q^{60}-2q^{58}+q^{54}-q^{52}-2q^{50}+2q^{46}-q^{42}-q^{40}+3q^{38}+3q^{36}+q^{34}-q^{32}+q^{30}+2q^{28}+2q^{26}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{86}-q^{84}+q^{82}-q^{80}+2q^{78}-2q^{76}+q^{74}-q^{72}+2q^{70}-q^{66}-2q^{62}+q^{60}-4q^{58}-4q^{54}+2q^{52}-3q^{50}+2q^{48}-3q^{46}+q^{44}+3q^{34}+q^{32}+4q^{30}+q^{28}+4q^{26}+q^{24}+2q^{22}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{148}-q^{146}+2q^{144}-2q^{142}+q^{138}-2q^{136}+5q^{134}-5q^{132}+4q^{130}-2q^{128}-3q^{126}+4q^{124}-6q^{122}+5q^{120}-3q^{118}-q^{116}+3q^{114}-3q^{112}+q^{110}+2q^{108}-5q^{106}+4q^{104}-3q^{102}-2q^{100}+5q^{98}-7q^{96}+8q^{94}-7q^{92}+2q^{90}+2q^{88}-6q^{86}+6q^{84}-7q^{82}+4q^{80}-2q^{76}+3q^{74}-3q^{72}+2q^{70}+3q^{68}-5q^{66}+3q^{64}-2q^{60}+7q^{58}-5q^{56}+5q^{54}-q^{52}+4q^{48}-4q^{46}+5q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}$

.

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["7 5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-4t+5-4t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}+4z^{2}+1$

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, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{8}\left(-z^{2}\right)-a^{8}+a^{6}z^{4}+2a^{6}z^{2}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{11}z^{3}-a^{11}z+2a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{8}z^{6}+a^{8}z^{2}-a^{8}+3a^{7}z^{5}-4a^{7}z^{3}+a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}-a^{5}z+a^{4}z^{4}-3a^{4}z^{2}+2a^{4}$

Vassiliev invariants

V2 and V3:

(4, -8)

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 $16$ $-64$ $128$ ${\frac {968}{3}}$ ${\frac {136}{3}}$ $-1024$ $-{\frac {5440}{3}}$ $-{\frac {928}{3}}$ $-224$ ${\frac {2048}{3}}$ $2048$ ${\frac {15488}{3}}$ ${\frac {2176}{3}}$ ${\frac {156422}{15}}$ ${\frac {5912}{15}}$ ${\frac {170888}{45}}$ ${\frac {730}{9}}$ ${\frac {7142}{15}}$

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 $s=$-4 is the signature of 7 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	χ
-3								1	1
-5							1	1	0
-7						2			2
-9					1	1			0
-11				2	2				0
-13			1	1					0
-15		1	2						-1
-17		1							1
-19	1								-1

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[7, 5]]
Out[2]=	7
In[3]:=	PD[Knot[7, 5]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 12, 6, 13], X[7, 14, 8, 1], X[13, 6, 14, 7], X[11, 8, 12, 9], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[7, 5]]
Out[4]=	GaussCode[-1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4]
In[5]:=	BR[Knot[7, 5]]
Out[5]=	BR[3, {-1, -1, -1, -1, -2, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[7, 5]][t]
Out[6]=	2 4 2 5 + -- - - - 4 t + 2 t 2 t t
In[7]:=	Conway[Knot[7, 5]][z]
Out[7]=	2 4 1 + 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 5], Knot[10, 130]}
In[9]:=	{KnotDet[Knot[7, 5]], KnotSignature[Knot[7, 5]]}
Out[9]=	{17, -4}
In[10]:=	J=Jones[Knot[7, 5]][q]
Out[10]=	-9 2 3 3 3 3 -3 -2 -q + -- - -- + -- - -- + -- - q + q 8 7 6 5 4 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 5]}
In[12]:=	A2Invariant[Knot[7, 5]][q]
Out[12]=	-28 -22 -18 -16 -14 -12 2 -6 -q - q - q + q + q + q + --- + q 10 q
In[13]:=	Kauffman[Knot[7, 5]][a, z]
Out[13]=	4 8 5 7 9 11 4 2 8 2 10 2 2 a - a - a z + a z + a z - a z - 3 a z + a z - 2 a z - 5 3 7 3 9 3 11 3 4 4 6 4 10 4 a z - 4 a z - 2 a z + a z + a z - a z + 2 a z + 5 5 7 5 9 5 6 6 8 6 a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[7, 5]], Vassiliev[3][Knot[7, 5]]}
Out[14]=	{0, -8}
In[15]:=	Kh[Knot[7, 5]][q, t]
Out[15]=	-5 -3 1 1 1 2 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 17 6 15 6 15 5 13 5 13 4 q t q t q t q t q t q t 2 2 1 1 2 1 ------ + ------ + ----- + ----- + ----- + ---- 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t